# Optimising hybrid rotational femtosecond/picosecond coherent anti-Stokes
Raman spectroscopy (HR-CARS) in nitrogen at high pressures and temperatures

Nils Torge Mecker  School of Engineering, The University of Edinburgh,
Edinburgh EH8 3JL, Scotland  Corresponding author<EMAIL_ADDRESS>Trevor
L. Courtney  Combustion Research Facility, Sandia National Laboratories,
Livermore, CA 94551, USA  Brian D. Patterson  Combustion Research Facility,
Sandia National Laboratories, Livermore, CA 94551, USA  David Escofet-Martin
School of Engineering, The University of Edinburgh, Edinburgh EH8 3JL,
Scotland  Brian Peterson  School of Engineering, The University of Edinburgh,
Edinburgh EH8 3JL, Scotland  Christopher J. Kliewer  Combustion Research
Facility, Sandia National Laboratories, Livermore, CA 94551, USA  Mark Linne
School of Engineering, The University of Edinburgh, Edinburgh EH8 3JL,
Scotland

###### Abstract

We demonstrate the use of hybrid rotational femtosecond/picosecond (fs/ps)
coherent anti-Stokes Raman spectroscopy (HR-CARS) as a technique for
temperature measurements in nitrogen gas at high pressures and temperatures. A
broadband pulse shaper-adjusted 42 fs pulse interacts with a narrow-bandwidth,
frequency-upconverted 5.5 ps pulse in a cell containing N2 at pressures of
1-70 atm and temperatures of 300-1000 K. A computational code is used to model
spectra and fit experimental results to obtain best-fit temperatures. We
demonstrate good qualitative fits as well as good accuracy and precision
between thermocouple measured and best-fit temperatures over the explored
pressure and temperature regimes. The overall average percentage temperature
difference between thermocouple measurements and best-fit temperatures is
-0.3% with a standard deviation of 7.1%, showing the suitability of HR-CARS
for characterising high pressure and temperature environments.

††journal: josab

##  1 Introduction

Coherent anti-Stokes Raman spectroscopy (CARS) was first used for gas-phase
measurements in the 1970s [1]. Since then, it has become a widely applied
technique for measurements of temperature and species concentration in flames
and other gas-phase applications [2]. CARS is a four-wave mixing technique
that combines three laser beams to create a fourth output beam which carries
the spectral information. It has several advantages over other methods such as
laser-induced fluorescence (LIF). These advantages include very good spectral
resolution and a laser-like directional signal [3]. CARS has been applied to a
variety of combustion processes at elevated temperatures and pressures, both
as vibrational CARS (VCARS) and also as pure rotational CARS (RCARS). VCARS
experiments in nitrogen (N2) have been reported for pressures $\leqslant$2500
atm, including temperature models and observations on pressure-induced
collisional narrowing [4, 5]. VCARS, however, suffers from a collapse of the
rotational structure at higher pressures, as the closely spaced ro-vibrational
lines start to mix [6]. In RCARS the line spacing is much bigger, resulting in
more highly resolved spectra, even at higher pressures. Studies of RCARS on
both N2 and O2 at various temperatures and pressures up to 150 atm have been
reported [7, 8, 9, 10].

The traditionally used nanosecond CARS has some important limitations,
including interference from the nonresonant (NR) background signal, low
acquisition rates and dependence on collisional dephasing. Picosecond CARS
overcomes interference from the nonresonant background as the probe beam can
be delayed in time from the pump and Stokes beams to acquire signal at a time
when the NR contribution has decayed significantly, while the CARS signal
remains strong. Unfortunately, collisional dephasing and low acquisition rates
remain limitations of picosecond CARS, similar to nanosecond CARS. These
problems were overcome by using femtosecond CARS, a technique that enables
collision-free measurements at high repetition rates. However, while ns and ps
CARS achieve high spectral resolution, fs CARS measures spectra in the time
domain, as the pulses are very short (10-100 fs), but accordingly broad in the
spectral domain. In 2006, hybrid femtosecond/picosecond (fs/ps) CARS was
developed by Prince et al. [11]. By using broadband femtosecond laser pulses
to create the Raman coherences and by probing them with a narrow-band
picosecond pulse, it combines the advantages of ps and fs CARS. The NR
background is avoided, while high repetition rates and high peak power can be
achieved, making single shot frequency domain measurements possible. In
addition, collisional dephasing can be avoided, at least at lower pressures.
For example, at 300 K and 60 atm, the collisional dephasing time is about 15
ps. In the two-beam hybrid fs/ps RCARS approach, a single femtosecond beam is
used as the pump and Stokes beams to excite the rotational Raman coherences.
This provides even further advantages, as it simplifies the experimental setup
and reduces the effect of beam steering, which is often encountered in
turbulent conditions [12].

Short-pulse CARS techniques have only been tested in a few experiments at
higher pressures. Papers have been published for moderate pressures at room
temperature in the time domain and up to 900 K in the frequency domain, using
fs-CARS and hybrid fs/ps CARS [13, 14, 15, 16, 17]. In this paper we present
an experimental setup for two-beam hybrid rotational fs/ps CARS (HR-CARS) of
N2 at pressures up to 70 atm and temperatures up to 1000 K. The research
presented here is a significant extension of a previous paper [18], including
a detailed presentation over the entire range of pressures and temperatures.
Our setup includes second harmonic bandwidth compression (SHBC) to create the
picosecond probe beam [19] and a pulse shaper to create a transform-limited
(TL) femtosecond pump/Stokes beam at the interaction volume. Further, we
present a computational model to generate theoretical HR-CARS spectra and we
validate it by fitting the experimental spectra to obtain best-fit
temperatures, thereby showing the usefulness of this technique and potential
use for thermometry in high pressure combustion environments in the future.

##  2 Theory and Model

Quantum mechanics forms the theoretical basis of CARS spectroscopy. In a semi-
classical treatment, using the density matrix and the Schrödinger equation
together with a classical electromagnetic optical wave, the 3rd order time-
domain polarisation, upon which the CARS signal depends, can be derived [20,
21]:

$\displaystyle P^{(3)}_{CARS}(t)=$
$\displaystyle\left(\frac{i}{\hbar}\right)^{3}\int_{0}^{\infty}dt_{3}\int_{0}^{\infty}dt_{2}\int_{0}^{\infty}dt_{1}R_{4}(t_{3},t_{2},t_{1})$
$\displaystyle E_{3}(t-t_{3})E_{2}(t-t_{3}-t_{2})E_{1}(t-t_{3}-t_{2}-t_{1})$
(1)

$E_{1}$, $E_{2}$ and $E_{3}$ are the time domain electric fields of the pump,
Stokes and probe beams respectively. $R_{4}$ is the 3rd order molecular
response function and $t_{1}$, $t_{2}$ and $t_{3}$ represent coherence
timescales. This nonlinear four-wave mixing process is depicted in Figure 1.

Figure 1: Schematic diagram of the four-wave mixing CARS process, showing the
interaction between the pump, Stokes and probe photons to produce a CARS
signal photon. Solid lines represent real molecular states (rotational states
in the case of RCARS) and dashed lines represent virtual states.

The 3rd order polarisation can be simplified, by assuming that the molecular
response is fast compared to the pump and probe pulse coherence time scales,
$t_{1}$ and $t_{3}$. This assumption is valid as the pump and probe pulses are
detuned from any direct one-photon electronic transition, so that molecular
dephasing over the $t_{1}$ and $t_{3}$ timescales is almost instantaneous. The
3rd order response function can then be replaced by
$R_{4}(t_{3},t_{2},t_{1})\rightarrow\delta(t_{3})R_{4}(0,t_{2},0)\delta(t_{1})$
[22, 23]. A further simplification is the assumption that the pump and Stokes
pulses are short and impulsive compared to the probe pulse, so that the pump
and Stokes electric fields can be replaced by delta functions. This is a valid
assumption for HR-CARS, as the picosecond probe is about two orders of
magnitude longer than the femtosecond pump/Stokes beam. Under these
assumptions, the 3rd order polarisation is given by:

$\displaystyle
P^{(3)}_{CARS}(t,\tau)=\left(\frac{i}{\hbar}\right)^{3}E_{pr}(t-\tau)R_{CARS}(t)e^{i\omega_{pr}(t-\tau)}$
(2)

where $E_{pr}$ is the electric field envelope of the probe beam with centre
frequency $\omega_{pr}$ and time delay $\tau$ relative to the pump/Stokes
beams. The molecular response function is denoted as $R_{CARS}(t)$ now. The
3rd order polarisation in the frequency-domain is given by a Fourier transform
of Equation 2. The frequency-domain CARS signal simply is the square of the
absolute value of the frequency domain 3rd order polarisation:

$\displaystyle
P^{(3)}_{CARS}(\omega,\tau)=\mathscr{F}\left[P^{(3)}_{CARS}(t,\tau)\right]$
(3) $\displaystyle
I_{CARS}(\omega,\tau)\propto\absolutevalue{P^{(3)}_{CARS}(\omega,\tau)}^{2}$
(4)

Equations 2, 3 and 4 are the underlying mathematics used in our computational
code. The 3rd order molecular response function, $R_{CARS}(t)$, is given by
the following phenomenological expression for S-branch rotational transitions
[22]:

$\displaystyle R_{CARS}(t)=$ $\displaystyle\sum_{v}\sum_{J}I_{v,J;v,J+2}$
$\displaystyle\times exp\left[\frac{t}{\hbar}\left(i\Delta
E_{v,J;v,J+2}-\frac{1}{2}\Gamma^{S}_{v,J;v,J+2}\right)\right]$ (5)

where the sum is performed over rotational states $J$ at each vibrational
level $v$. $\Delta E_{v,J;v,J+2}$ and $\Gamma^{S}_{v,J;v,J+2}$ are the
frequency and linewidth of a transition between rotational states $J$ and
$J+2$ at vibrational level $v$ respectively. $I_{v,J;v,J+2}$ is the
corresponding Raman transition strength. For each individual transition,
Equation 5 is a sinusoidal curve with a frequency based on the transition
frequency ($\Delta E_{v,J;v,J+2}$) which decays exponentially with linewidth
($\Gamma^{S}_{v,J;v,J+2}$) and has an amplitude given by the transition
strength ($I_{v,J;v,J+2}$). The sum over all rotational states at each
vibrational level results in a time-domain Raman signal. The individual
exponentially decaying sinusoidal waves are in phase at certain times and out
of phase at others, resulting in a periodic structure as shown in Figure 2.
This is because as the rotational coherences set up by the pump/Stokes beams
evolve in time they become dephased and then rephased again in certain
intervals. Whenever they are rephased there is a peak in the Raman signal. For
rotations, rephasing occurs in intervals of
$\tau_{full}=\nicefrac{{1}}{{(2B_{e}c)}}$, where $c$ is the speed of light and
$B_{e}$ the equilibrium rotational constant. The first full revival of N2
($B_{e}$=1.99826 cm-1 [24]) is at 8.4 ps. Partial rephasing occurs at
$\nicefrac{{1}}{{4}}\ \tau_{full}$, $\nicefrac{{1}}{{2}}\ \tau_{full}$ and
$\nicefrac{{3}}{{4}}\ \tau_{full}$.

Figure 2: Real part of the molecular response function of N2 in the time
domain for pure rotational S-branch transitions at 293 K and 1 atm. The first
full revival is at 8.4 ps (and the second at 16.7 ps). Partial rephasing
occurs at $\nicefrac{{1}}{{4}}\ \tau_{full}$, $\nicefrac{{1}}{{2}}\
\tau_{full}$ and $\nicefrac{{3}}{{4}}\ \tau_{full}$.

The Raman transition strength is given by the following [22]:

$\displaystyle
I_{v,J;v,J+2}\propto\frac{4}{45}b_{J,J+2}\times(\gamma^{\prime}_{v})^{2}\times
F_{rot}(J)\times\Delta\rho_{v,J;v,J+2}$ (6)

where $b_{J,J+2}$ is the Placzek-Teller Coefficient, $\gamma^{\prime}_{v}$ is
the polarisation anisotropy, $F_{rot}(J)$ is the Herman-Wallis Factor and
$\Delta\rho_{v,J;v,J+2}$ is a population difference factor, based on the
Boltzmann distribution of the rotational states.

The Placzek-Teller Coefficient, first described by Placzek and Teller in 1933
[25], determines the dependence of the Raman transition strength on the
rotational level. As a molecule rotates faster (i.e. as its rotational quantum
number increases), its internuclear separation gets larger, changing its
moment of inertia. This centrifugal distortion changes the polarisability of
the molecule, which in turn changes the Raman transition strength. For
S-branch transitions of linear diatomic molecules, this factor is:

$\displaystyle b_{J,J+2}=\frac{3(J+1)(J+2)}{2(2J+1)(2J+3)}$ (7)

Similar to the change in polarisability (and hence the Raman transition
strength) due to the rotation of the molecule, there is also a change in
polarisability due to the interaction between rotations and vibrations. This
interaction arises from there being many vibrations during a single rotation.
The change in Raman transition strength due to this ro-vibrational interaction
is described by the Herman-Wallis Factor. For S-branch rotational transitions
of nitrogen, there are different versions of the Herman-Wallis Factor in the
literature [26]. Our model uses the S-branch Herman-Wallis Factor by Tipping
and Ogilvie [27], which is also used in other models [22].

$\displaystyle
F_{rot}(J)=\left[1+\kappa^{2}\left(\frac{p_{1}}{p_{0}}\right)(J^{2}+3J+3)\right]^{2}$
(8)

where $\kappa=\nicefrac{{2B_{e}}}{{\omega_{e}}}=1.695\times 10^{-3}$ for N2
with vibrational frequency $\omega_{e}$ = 2358.518 cm-1 [24]. The ratio
between the first two coefficients of the anisotropic polarisability expansion
is $\nicefrac{{p_{1}}}{{p_{0}}}=3.168$ [28].

The polarisation anisotropy is dependent on the vibrational level according to
[29]:

$\displaystyle\gamma^{\prime}_{v}=\beta_{e}+\beta_{e}^{\prime}r_{e}\left[\frac{3B_{e}}{\omega_{e}}+\frac{\alpha_{e}}{2B_{e}}\right]\left(v+\frac{1}{2}\right)$
(9)

where for N2, $\beta_{e}=6.91\times 10^{-25}$ cm3 is the equilibrium
polarisation anisotropy, $\beta_{e}^{\prime}=1.4\times 10^{-16}$ cm2 is the
first derivative of the polarisation anisotropy with respect to the
internuclear distance, $r_{e}=1.098\times 10^{-8}$ cm is the equilibrium
internuclear distance and $\alpha_{e}=1.7305\times 10^{-2}$ cm-1 [24] is the
first rotational-vibrational interaction constant.

Finally, the population difference factor is calculated from a normalised
Boltzmann distribution of rotational and vibrational states [22]:

$\displaystyle\Delta\rho_{v,J;v,J+2}$
$\displaystyle=N_{v,J}-\frac{2J+1}{2J+5}N_{v,J+2}$ $\displaystyle N_{v,J}$
$\displaystyle=\frac{g_{J}(2J+1)}{Z}exp\left(\frac{-F_{v,J}}{k_{B}T}\right)$
(10)

where $Z$ is the partition function. For N2, if $J$ is even, the rotational-
nuclear degeneracy is $g_{J}=6$ and if $J$ is odd, it is $g_{J}=3$. $F_{v,J}$
is the rotational energy of rotational state $J$ in vibrational level $v$.
This energy is calculated based on the non-rigid rotor approximation (i.e.
including centrifugal distortion) and taking into account the interaction
between rotations and vibrations of the molecule [30]:

$\displaystyle F_{v,J}$
$\displaystyle=\omega_{e}\left(v+\frac{1}{2}\right)-\omega_{e}\chi_{e}\left(v+\frac{1}{2}\right)^{2}+BJ(J+1)-DJ(J+1)^{2}$
$\displaystyle B$
$\displaystyle=B_{e}-\alpha_{e}\left(v+\frac{1}{2}\right)+\gamma_{e}\left(v+\frac{1}{2}\right)^{2}$
$\displaystyle D$ $\displaystyle=D_{e}+\beta_{e}\left(v+\frac{1}{2}\right)$
(11)

where for N2, $D_{e}=5.774\times 10^{-6}$ cm-1 and
$\omega_{e}\chi_{e}=14.2935$ cm-1 are the equilibrium centrifugal constant and
second order vibrational constant respectively. $\alpha_{e}=1.73035\times
10^{-2}$ cm-1, $\beta_{e}=1.55\times 10^{-8}$ cm-1 and
$\gamma_{e}=-3.1536099\times 10^{-5}$ cm-1 are the rotational-vibrational
interaction constants [24]. In our model the ground state ($v=0$) and first
excited state ($v=1$) vibrational levels are included, as 3.5% of the
molecules are in the first excited vibrational state at 1000 K.

The transition frequencies for the S-branch, $\Delta E_{v,J;v,J+2}$, i.e. the
energy differences between rotational levels $J+2$ and $J$ in vibrational
level $v$, are calculated from Equation 11 according to the selection rules
$\Delta v=0$ and $\Delta J=+2$ for S-branch transitions [31]:

$\displaystyle\Delta E_{v,J;v,J+2}$ $\displaystyle=F(v,J+2)-F(v,J)$
$\displaystyle=2B(2J+3)-4D(2J+3)(J^{2}+3J+3)$ (12)

The transition linewidths, $\Gamma_{v,J;v,J+2}$, can be calculated from
linewidths models, such as the modified exponential gap (MEG) model. Steinfeld
et al. [32] provide an overview of many of these linewidths models. However,
the linewidths used in our computational code are based on interpolated
experimental measurements by Kliewer et al. [33].

The probe electric field in Equation 2, $E_{pr}(t-\tau)$, is modelled as the
sum of Gaussians, including a chirp factor [34]:

$\displaystyle E_{pr}(t-\tau)$ $\displaystyle=E_{probe}(t-\tau)\times
E_{chirp}(t-\tau)$ $\displaystyle E_{probe}(t-\tau)$
$\displaystyle=\sum_{j}A_{j}exp\left[-\frac{2ln(2)(t-\tau_{j})^{2}}{\tau^{2}_{j}}\right]$
$\displaystyle E_{chirp}(t-\tau)$
$\displaystyle=exp\left[-i\frac{2ln(2)\alpha(t-\tau)^{2}}{\Delta
t_{exp}^{2}}\right]$ (13)

A sum of two Gaussians ($j=2$) is chosen to fit measured cross-correlations of
the probe beam in argon, which is not a perfect Gaussian, but shows
significant asymmetry. Careful modelling of the probe pulse shape is required
for accurate CARS spectra. The chirp factor, $\alpha$, is included in our
model because we measured the bandwidth of the probe to be $\nu=3.0$ cm-1.
This bandwidth has a transform limit of 4.9 ps, but cross correlations in
argon gave a pulse duration, $\Delta t_{exp}$, of 5.5 ps, 12% longer than the
TL. Including the chirp stretches the probe from its TL to the required
length. Without this chirp factor, the model has narrower peaks and shifted
frequencies when compared with experimental spectra. Further, including the
chirp allows better modelling of spectrally overlapping transitions that can
be seen in our experimental data. Figure 3 shows a measured cross correlation
of the probe beam, along with a two-Gaussian fit.

Figure 3: Probe pulse in the time domain ($E_{probe}=\sqrt{I_{probe}}$) as
measured by a cross-correlation (cross corr.) in argon (blue line). A two-
Gaussian probe pulse (red line) as modelled by Equation 13 is shown as well.

The final step in modelling CARS spectra is to convolve the spectrum with an
instrument slit function, modelled as a 1.5 cm-1 Gaussian in our case. We
determined this function by measuring the spectral linewidths of mercury
emission lines near the probe frequency.

Figure 4 shows model results for two different conditions. Graph (a) shows the
molecular response function, the probe pulse shape and the resulting CARS
signal at 1 atm of N2, while (b) shows the same at a pressure of 69 atm.
Looking at the molecular response (grey line, same as Figure 2, but as an
absolute value), one can see that it decays very slowly at low pressure, but
at high pressure it drops to less than a tenth within the first 10 ps, as the
decay is exponential with pressure. However, there remains a significant CARS
signal (blue line) even at high pressure, because the short picosecond probe
(black line) overlaps with most of the molecular response. Additionally, the
probe is not present at time zero, so it is not necessary to include the
nonresonant signal in the model.

Figure 4: Time domain model of the absolute value of the real part of the
molecular response (same as Figure 2, but as an absolute value, grey line),
the two-Gaussian probe pulse shape (black line) and the real part of the
corresponding CARS signal (blue line). (a) displays these functions at 1 atm
and (b) at 69 atm. In both cases, the model is for pure N2 at 293 K and the
peak of the probe pulse is centred at a time delay of 8.4 ps, the first full
revival of the molecular response of N2. The probe pulse shape shown here is
an example, based on one of the measured cross correlations.

In the frequency domain, the CARS spectra corresponding to the low and high
pressure time-domain CARS signals of Figure 4 are shown in Figure 5. It can be
seen that even at high pressure the lines are relatively narrow and well
resolved, because the probe overlaps significantly with the molecular
response. The non-centred shapes of the small lines (that arise from
transitions involving odd rotational states) arise from the non-Gaussian shape
of the probe pulse in combination with the chirp.

Figure 5: Frequency domain N2 HR-CARS spectra at 293 K and 1 atm (blue line)
and 69 atm (red line), corresponding to the low and high pressure CARS signals
in Figure 4. A 1.5 cm-1 slit function has been convolved with the spectra.

Our computational code can model the pump/Stokes beams explicitly, without
assuming impulsive excitation. However, we found that the difference between
the explicit and impulsive model is marginal, at least for the pump/Stokes
beams used in our experiments. We therefore modelled our spectra with
impulsive excitation as this is computationally faster. Modelling of the
nonresonant response is unnecessary, as the spectra presented in this paper
were delayed long enough so that no significant NR signal was present. The
G-matrix, which accounts for line-mixing at very high pressures, was not
included in the model. This is because according to our calculations, line
mixing does not have a significant effect on rotational transitions, even at
the highest pressures investigated (69 atm).

##  3 Experimental Setup

The experimental setup described here is the same as the one reported in a
previous paper [18]. Both, the pump/Stokes and the probe beams, come from the
same laser system, a Coherent Legend Elite Ti:Sapphire regenerative amplifier
(790 nm, 1 kHz), that is seeded by a Coherent Vitara oscillator (100 MHz). The
output is split into two beams of equal power before being single pass
amplified to about 9 mJ/pulse. These resulting 795 nm broadband beams have a
width of $\Delta\nu$ = 360 cm-1 and are strongly positively chirped ($>$100
ps). One of these output beams is used to create the femtosecond pump/Stokes
beam and the other one is used to create the picosecond probe beam. A
schematic diagram of the entire HR-CARS experimental setup is given in Figure
6.

Figure 6: Two beam HR-CARS experimental setup. The beam from a 1 kHz
Ti:Sapphire laser system is split into two equal power beams. The pump/Stokes
pulse is generated by grating-compressing one of the output beams followed by
dispersion compensation in a pulse shaper that uses a spatial light modulator
(SLM). The probe is generated using second harmonic bandwidth compression
(SHBC), by splitting the other amplifier output in two, chirping the two beams
in grating compressors equally and opposite and combining them in a beta
barium borate (BBO) crystal. Spectral wings are filtered in a folded 4f
grating filter. The pump/Stokes and probe beams are combined collinearly and
focussed into a pressure cell. The resulting CARS signal is filtered in a
folded 4f grating filter and focussed into a spectrometer. R: reflection, T:
transmission, SL: spherical lens, CL: cylindrical lens, G: grating, CM: curved
mirror, DM: dichroic mirror, SP: short pulse, BS: beam splitter, ND: neutral
density.

The pump/Stokes beam is created by using one of the stretched and amplified
output beams and compressing it in a grating compressor to its near-transform
limit of about 40 fs. Using a series of beam splitters, the energy of the beam
is reduced to 35 $\mu$J/pulse. Furthermore, the beam is down-collimated using
300 mm and 100 mm spherical lenses, from a $\nicefrac{{1}}{{e^{2}}}$ beam
diameter of 12 mm to 4 mm. The beam then enters a pulse shaper (Femtojock).
The reduction in power is dictated by the spatial light modulator (SLM) in the
pulse shaper. A diagram of the pulse shaper is shown in Figure 7. It uses a 4f
design to separate the different colours of the beam with a grating and
focusses them with a curved mirror onto an SLM at the Fourier plane. The SLM
is a liquid crystal array of pixels, to each of which a different voltage can
be applied to create a relative delay of the different colours of the beam.
The so-called multiphoton intrapulse interference phase scan (MIIPS) algorithm
[35, 36] is used to create a compensation phase mask to be applied to the SLM
to account for dispersion in the pump/Stokes beam, mainly induced from the 28
mm high pressure cell fused silica entrance window. Linear and nonlinear
dispersion are accounted for using this setup, creating a nearly transform-
limited pulse at the interaction probe volume inside the pressure cell.

Figure 7: 4f pulse shaper. Light is dispersed in a grating and the different
colours are focussed with a curved mirror onto a spatial light modulator
(SLM), that is located at the Fourier plane. The liquid crystal SLM array can
delay the different colours by different amounts of time, creating a nearly
dispersion free pulse. CM=curved mirror.

Frequency-resolved optical gating (FROG) [37, 38] was used to measure the
pulse to be 42 fs with an almost constant phase over the length of the pulse
after passing through the pressure cell window, as shown in Figure 8. The
energy of the beam after the pulse shaper is 12 $\mu$J/pulse, about a third of
the input energy.

Figure 8: FROG measured intensity (blue), transform limit (red) and phase
(green) of the pump/Stokes pulse after passing first through the pulse shaper
and then a 28 mm quartz window. (a) without a compensation phase mask: the
intensity is far from its TL and has a width (FWHM) of 112 fs; the phase
changes significantly over the length of the pulse. (b) with a compensation
phase mask: the intensity is almost at its TL and has a width of 42 fs; the
phase is almost constant over the length of the pulse.

Using the mask significantly increases the excitation efficiency of the
pump/Stokes beam (also denoted the experimental spectral response). This can
be seen in Figure 9, which shows two nonresonant argon CARS spectra, one
recorded with the mask and one without. The excitation efficiency with the
mask is better, as there is an increased NR response across all wavenumbers,
compared to not using a mask. Especially at higher wavenumbers there remains a
relatively large response when the mask is used. This is because pump/Stokes
excitation at higher wavenumbers requires an increasingly larger frequency
separation between intrapulse pump and Stokes interactions, which quickly lose
time coincidence with chirp. NR argon spectra are used to correct experimental
N2 spectra for the pump/Stokes excitation efficiency.

Figure 9: Nonresonant CARS spectra of argon showing the pump/Stokes excitation
efficiency. One is taken without the phase mask on the SLM (red line),
corresponding to Figure 8 (a) and one is taken with the phase mask on the SLM
(blue line), corresponding to Figure 8 (b).

The probe beam is generated from the other amplifier output beam via sum-
frequency generation (SFG) using a second harmonic bandwidth compression
(SHBC) method. This setup, adapted from a technique by Raoult et al. [39], is
described in full detail in a different paper [19]. While most SHBC setups
start with compressed beams [34, 40], our design feeds the uncompressed
amplifier output to the SHBC setup, as can be seen in Figure 6. This reduces
the number of gratings needed in the setup, making it simpler, while at the
same time leading to a higher power throughput. The uncompressed strongly
chirped amplifier output is split into two equally powerful beams, that are
each routed through a grating compressor which have their gratings shifted in
an equal but opposite way from the separation needed for complete pulse
compression, thereby chirping the two beams equally but opposite. The two
chirped beams are then down collimated with 500 mm and 250 mm spherical lenses
from a beam size of 10 mm to 5 mm. This increases the energy density and
allows for more energy-efficient sum-frequency generation, without damaging
the crystal, as would happen if the beams were focused onto it. The two beams
are combined in a 1 mm beta barium borate (BBO) crystal (Type1), in which
their frequencies add via SFG to produce a narrowband unchirped picosecond
pulse centred at 396.5 nm. However, in reality the chirp of the fundamental
beams has nonlinear temporal contributions, resulting in a time-dependent SFG
beam that is slightly blue-shifted from its minimum frequency at the edges of
the pulse [19].

The beams are crossed at a very narrow angle ($\sim 1^{\circ}$), to minimise
spatial chirp. The resulting beam has a full width at half maximum (FWHM)
bandwidth of 3 cm-1 (deconvolved with the spectrometer’s slit function,
separately measured to be 1.5 cm-1) and a duration of about 5.5 ps, as
confirmed with a cross-correlation of the NR signal in pure argon. Figure 10
shows these beam characteristics in the frequency and time domain. A
translation stage in one of the fundamental beams before the BBO crystal is
used to control the relative delay between the oppositely chirped pulses,
allowing for fine tuning of the SFG spectrum. Further, two stacked angle-tuned
short pass filters in the other fundamental beam are used to cut out the
leading edge of that pulse, resulting in an SFG pulse with a relatively clean
and steep early time profile, albeit not as steep as anticipated. This SHBC
setup has an energy conversion efficiency of 20% as the combined 6 mJ/pulse
from the input pulses result in a 1.2 mJ/pulse SFG beam. The SFG beam emerging
from the BBO crystal is filtered with a neutral density filter (1.0 ND) to
reduce its power to avoid damaging the grating of the grating filter (see
below) and collimated with 50 mm and 100 mm spherical lenses, to reduce
spatial variations in the beam intensity. At this point, the beam has spectral
wings arising from imperfect conjugate chirping (frequency distributions at
each point in time in the broadband pulses), together with blue-shifted
satellite peaks that are a result of interferences in the SFG of temporally
chirped fundamental pulses [19]. These are filtered out in a folded 4f grating
filter (f.l.=1 m, 1800 grooves/mm grating) with a slit at the Fourier plane.
The CARS signal lineshapes are significantly simplified when the satellite
pulses are filtered out. The spectral wings, though a lot weaker than the peak
of the probe, are similar in strength to the CARS signal. Their reduction
diminishes the background noise coming from the probe, that otherwise is
present in collinear CARS.

Figure 10: Probe beam as created by the SHBC setup described above. (a) in the
frequency domain measured in a spectrometer, FWHM of 3.5 cm-1 (3 cm-1 after
deconvolution with the spectrometer’s 1.5 cm-1 slit function). (b) in the time
domain as measured with a cross correlation in nonresonant argon.

The two beams are routed to a high pressure gas cell in which they are
combined collinearly through the use of a dichroic mirror that transmits the
396.5 nm probe beam and reflects the 800 nm pump/Stokes beam. Before
combination in the dichroic mirror, the polarisation of the pump/Stokes beam
is rotated by 90∘ in a periscope, so that the two beams have matched vertical
polarisations. Time delay between the pulses is provided by a computer-
controlled translation stage (ThorLabs NRT100). The beams are focussed into
the cell with an uncoated spherical lens of focal length 300 mm. The power of
the beams just before the cell is measured to be 10 $\mu$J/pulse for the
pump/Stokes pulse and 5 $\mu$J/pulse for the probe beam. The
$\nicefrac{{1}}{{e^{2}}}$ diameters at the beam waists are 100 $\mu$m and 70
$\mu$m respectively. The collinear geometry is chosen to achieve an increased
signal from the rather low power input beams by creating a longer CARS
interaction region, which is still much shorter than the length of the cell.
The custom-built pressure cell itself is an insulated steel cylinder of length
30 cm and diameter 15 cm, with the gas to be probed being contained in a core
of diameter 2.5 cm in the centre of the cylinder. Self-phase modulation of the
pump/Stokes beam that has been reported for other CARS experiments [41] was
not observed, due to the low power of the beam. After the cell, a 200 mm focal
length spherical lens is used to collimate the exiting pump/Stokes, probe and
signal beams. To filter the probe from the signal beam, a 4f grating filter
with a sharp edge at the Fourier plane is used. This ensures that the probe is
almost completely removed from the signal, while retaining all CARS
transitions, including even the $J=2$ lines, which are just 28 cm-1 from the
probe. Optical short-pass filters are not able to provide a sufficiently sharp
edge, which is why the grating filter is used. A 75 mm spherical lens is used
to focus the CARS signal onto the entrance of a 0.550 m Czerny-Turner
spectrometer (Horiba iHR550, 2400 grooves/mm grating). It is then dispersed
onto a back-illuminated CCD (Andor DU971N-BV, 400 $\times$ 1600 pixel, 16
$\mu$m pixel size). 1 kHz single-shot data collection is achieved by
vertically binning the CARS signal in an 11 $\times$ 1000 pixel region in $<$1
ms (10 $\mu$s exposure).

N2 HR-CARS spectra were measured over a large range of temperatures and
pressures. The high pressure stainless steel cell is rated for up to 70 atm at
room temperature, with the maximum pressure decreasing linearly with
increasing temperature, to 38 atm at 1000 K. This leads to a trapezoid of
temperatures and pressures (T-P) that can be achieved with this cell, as shown
in Figure 11, which also shows all the temperatures and pressures at which
spectra were taken. At each of those T-P points, several spectra were taken at
different time delays, all 1 ps apart and centred at about 8.4 ps, which is
the time of the first full revival of N2. Temperatures were measured with a
thermocouple inserted into a separate hole in the steel cylinder, about 5 cm
from the centre of the cell. The thermocouple measurements were validated by
separately testing the cell with an additional thermocouple that was placed in
the core of the cell, which holds the probed gas. It was found that the
readings for both thermocouples were in good agreement. To ensure that the gas
had the same temperature as the cell, the cell was kept sealed for some
minutes prior to measurement or kept at a minimum flow necessary for
maintaining the desired pressure. Nonresonant argon spectra were also
recorded, as part of cross correlations to determine the probe pulse shape in
time, and also to correct for the excitation efficiency of the pump/Stokes
beam. At each temperature and pressure, 1000 single laser shots were
collected.

Figure 11: Maximum and minimum temperatures and pressures for which the
pressure cell is rated (bound by the red trapezoid) and temperatures and
pressures at which HR-CARS spectra were recorded (blue dots).

##  4 Fitting Procedure

In preparation for fitting the experimental spectra to the computational
model, the 1000 single-shot spectra that were recorded at each temperature,
pressure and time delay were averaged and background subtracted. Further, they
were divided by a background-subtracted nonresonant argon spectrum to correct
for the pump/Stokes excitation efficiency. Finally, the experimental CARS
spectra were normalised. The argon spectra used for correction were taken on
the same day and at the same pressure as the CARS spectra. This is because we
observed narrowing of NR argon spectra with pressure and in addition, the NR
spectra changed with time, from day to day and even during the same day. This
is shown in Figure 12, which displays a clear trend in narrowing of NR spectra
with increasing pressure and also shows significant differences between NR
spectra taken at the same pressure but different times. The narrowing with
pressure can be explained by the increasing number density of molecules inside
the cell that the beam has to travel through before the probe volume and which
induces chirp. The variation of the nonresonant response with time is believed
to arise from the pulse shaper; slight alignment drift over time (common to
many laser systems) into the SLM can have large effects on the dispersion of
the pump/Stokes pulse after passing through the pressure cell window, which in
turn has a big effect on the excitation efficiency. This shows how important
it is to measure NR argon spectra close in time to the real CARS spectra and
at the same pressure, as otherwise the temperature fits will be off by a
significant amount if a wrong pump/Stokes excitation efficiency correction is
used.

Figure 12: Nonresonant CARS spectra of argon. (a) displays spectra taken at
the same time but different pressures. (b) displays spectra taken at the same
pressure (21 atm) but at different times during the same day. The time elapsed
between spectrum #1 and #2 is 2:18 h and between #2 and #3 is 1:41 h.

To obtain a wavenumber axis for the experimental spectra, a linear fit of
position of the experimental CARS peaks in pixel space to their theoretical
position in wavelength space is made using a low pressure CARS spectrum with a
time delay close to the first revival of N2, because of its simplicity and
high resolution. For this, the theoretical transition frequencies are
calculated in wavenumbers. The wavenumber of the probe beam (25221 cm-1) is
added and the result is converted to wavelength. These numbers are then fit
linearly to their positions in pixel space. This leads to a linear conversion
equation of the form $wavelength=m\times pixel+c$, where $m$ is the gradient
and $c$ the intercept. To represent experimental spectra on a wavenumber axis,
the calibrated wavelength axis is inverted to wavenumber and the wavenumber of
the probe is subtracted. The gradient and intercept values for different
spectra at different temperatures were all very close, on average
$m=-0.009569$ nm/pixel (standard deviation of 7.2$\times$10-6) and $c=396.238$
nm (standard deviation of 0.016) respectively. Regression statistics give an
average R2 value of 0.99999 showing that a linear wavelength calibration is
satisfactory.

Experimental spectra are then fit to the computational spectra through a non-
linear least squares fitting routine in MATLAB called lsqcurvefit. This allows
one to define fitting parameters and to place lower and upper constraints on
those parameters. In our fitting, the wavenumber axis was allowed to slightly
shift and stretch by allowing the gradient ($m$) and intercept ($c$) of the
linear wavelength calibration to vary within three standard deviations. The
relative height of the model to the experimental spectra was varied between
95% and 105% to account for a possible intensity offset of the highest peak.
Because of the non-Gaussian shape of the probe pulse and the resulting
uncertainty in exact probe pulse delay, the temporal centre of the probe pulse
was allowed to vary within $\pm$2 ps of the experimentally verified delay
based on the corresponding cross correlation. This was done for the most
resolved spectrum at each temperature and pressure. The time delays for the
other spectra at a given temperature and pressure are then steps of 1 ps
$\pm$0.2 ps from the best fit probe delay of the most resolved spectrum.
Similarly, the chirp parameter, $\alpha$ in Equation 13, was floated between 0
and -2.0 for the most resolved spectrum and its best fit value $\pm$0.2 was
used for the other time delays at the same temperature and pressure. Finally,
the temperature was allowed to vary within $\pm$30% from the thermocouple
temperature reading recorded for a given spectrum. The pressure was kept fixed
at its measured value.

All spectra were fit according to this procedure. Six fits of spectra taken at
very different temperatures and pressures are shown as an example in Figure
13. The spectra shown were all taken at a probe delay of 8 ps, apart from the
highest pressure one, (c), which was taken at a delay of 11 ps, as at this
pressure the earlier probe time delay spectra have a slight nonresonant
contribution due to the resonant CARS signal being relatively low at very high
pressure. It can be seen that across all the different temperatures and
pressures fits are generally very good, as confirmed by the low residuals.
Further, the best fit temperatures of the shown spectra are accurate, the
percentage differences between best fit temperatures and thermocouple readings
(as a percentage of the thermocouple reading) range from -6.2% to 2.3%.

Figure 13: Experimental and modelled HR-CARS spectra of N2. Experimental (blue
line) and best-fit model spectra (dashed red line) are shown together with the
absolute value of the residual (experiment-model, green line). The spectra
were all taken at a probe delay of 8 ps, apart from (c), which was taken at a
delay of 11 ps. The best fit temperatures and percentage differences to the
corresponding thermocouple measurements (as a percentage of the thermocouple
measurement) are: (a) 341 K, 2.3%; (b) 322 K, -3.2%; (c) 312 K, -6.2%; (d) 685
K, 0.4%; (e) 980 K, 1.1%; (f) 955 K, -1.8%.

A study of the sensitivity of the fits on the chirp and probe delay was
performed on four spectra from the four corners of the T-P trapezoid (Figure
11) at 333 K and 8 atm, 333K and 67 atm, 1000 K and 8 atm, 1000 K and 37 atm.
This sensitivity study is not statistically representative of all spectra
taken, but should serve as a strong indication as to how much the chirp and
time delay affect the best fit. For this study, the chirp and probe delay were
changed in turn, while all the other fitting parameters were kept at their
best fit value, as found by the afore described fitting procedure. The chirp
parameter, $\alpha$, was offset by $\pm$0.2 and $\pm$0.5 from its best fit
value and the probe time delay was offset by $\pm$0.2 ps and $\pm$1.0 ps. In
each of these cases, the residual and temperature fit were recorded. Figure 14
shows the results as a percentage difference of the residual and temperature,
relative to their best fit values.

Figure 14: Sensitivity study of probe delay and chirp. (a) and (b) show the
percentage differences in residual as compared to the best fit residual
(residual diff.) for different probe delay and chirp offsets from their best
fit. (c) and (d) show the corresponding percentage differences in temperature
as compared to the best fit temperature (temp diff).

For both, chirp and probe delay, it can be seen that there is a significant
increase in residual when these two parameters are changed from their best fit
value, especially when the change is large but still within their uncertainty.
However, the temperature fits do not follow this trend. For all tested spectra
apart from the highest pressure one (67 atm) the temperature fits only change
by a few percent, even if the corresponding residual increases by more than
100%. Only the highest pressure spectrum shows changes of more than 10% in the
temperature fits. Overall, varying the fitting parameters significantly
increases the residual, but does not affect the temperature fits
significantly, except for the highest pressure spectra. This shows that the
high pressure spectra need to be modelled very carefully to get a good fit
with the experiment.

##  5 Best Fit Temperatures

All spectra taken at the various pressures, temperatures and time delays were
fit using the afore described fitting procedure. Across all spectra, the
average percentage temperature difference of the best fit temperature to the
thermocouple temperature (as a percentage of the thermocouple reading) is
-0.3% with a standard deviation of 7.1%. This means that 84% of the fits fall
within $\pm$10% of the measured thermocouple temperature. The average absolute
percentage temperature difference (opposed to the average percentage
temperature difference, which averages positive and negative percentage
temperature differences) is 5.3%.

We focus a statistical analysis on the data taken at a time delay of 8 ps, the
time delay which is closest to the first full revival of N2 and generally
results in the most well resolved spectra. Best fit temperature differences to
the thermocouple reading at all the tested pressures and temperatures for this
time delay are shown in Figure 15. Each line in this graph represents a
different pressure. Note that the high pressure line contains the data for the
highest pressure measured at each temperature, starting at 69 atm at 293 K and
decreasing linearly to 37 atm at 1000 K, following the inclined edge of the
T-P trapezoid (Figure 11). Similarly, the low pressure data follow the left
edge of the T-P trapezoid. At low pressure and 8 atm the results are most
accurate, with more than 95% of the data lying between $\pm$10% of the
measured thermocouple temperatures. For medium pressures (21 atm and 35 atm)
that number drops to 73% and 87% respectively. But even at the highest
pressures, 78% of the data lie within $\pm$10% of the thermocouple
measurement, indicating a good accuracy.

Figure 15: Plots of % temperature difference (temp. diff.) between the
measured thermocouple temperature and the best fit temperature vs. absolute
measured temperature. Each line shows data at a different pressure. The black
lines, shown for reference, indicate the $\pm$10% range. See text for
definition of _low P_ and _high P_.

It should be noted that the low pressure best fit temperatures (Figure 15,
blue line) mostly lie within $\pm$5% from the thermocouple temperature, not
quite as accurate as previously reported accuracies, which are about 1% for
rotational N2 spectra at atmospheric pressure [42]. This is because the
experimental setup is optimised for high pressure CARS experiments, and some
of its components (e.g. the high pressure cell windows and the pulse shaper)
add complexity to the setup making the experimental conditions less ideal than
they could be for purely low pressure CARS studies. However, this complexity
is considered necessary for the acquisition of well-resolved and relatively
strong high pressure CARS spectra, thereby helping to improve the accuracy at
higher pressures.

To get a better overall understanding of the best fit data, a plot of all the
fits at 8 ps probe delay is shown on a T-P trapezoid, in Figure 16. The fits
are grouped such that all data within the same 5% accuracy interval have the
same colour. It can be seen that the majority of the fits are of high
accuracy, across all temperatures and pressures. Especially at low pressures,
the fits are all very accurate, the same is also true for fits at higher
temperatures across all pressures. It is only at the higher pressures at low
temperatures that some fits are less accurate than desired.

Figure 16: Percentage temperature difference (t.d.) for 8 ps probe delay
spectra on a temperature-pressure graph. The fits are grouped such that all
data within the same 5% accuracy interval (percentage difference between best
fit and thermocouple-measured temperature) have the same colour.

When looking at the T-P trapezoid of all the different time delays, the
results look similar (Figure 17). At low pressures (up to 8 atm) the fits are
all very accurate, across all temperatures. Also at medium pressures (between
20 and 40 atm), most fits are within $\pm$10% of the measured thermocouple
temperature, especially at higher temperatures. Even at the highest pressures
(above 50 atm), most fits are still of acceptable accuracy. However, in this
pressure regime, there are also a few fits that are less accurate. On average,
however, the accuracy is good both a low and at high pressures, while the
precision is a lot better at lower temperatures. At 8 atm, the average
percentage difference between the best fit temperature and the measured
thermocouple temperature, across all temperatures and time delays is 0.4%
(absolute average: 3.0%) with a standard deviation of just 3.9%. For the high
pressure data, the average is a relatively small -2.2% (absolute average:
7.0%) with a larger standard deviation of 8.4%.

Figure 17: Percentage temperature difference (t.d.) of all recorded spectra on
a temperature-pressure graph. The fits are grouped such that all data within
the same 5% accuracy interval (percentage difference between best fit and
thermocouple-measured temperature) have the same colour.

Some of the larger deviations of the best fit to the measured thermocouple
temperatures can be explained by a drift of the pump/Stokes beam pointing and
dispersion with time. This especially affects the effectiveness of the pulse
shaper to create an almost TL beam. The usefulness of the pulse shaper to
create an almost TL pulse after passing through the window of the pressure
cell (Figure 8), and therefore to generate a larger pump/Stokes excitation
bandwidth (Figure 9) has been discussed above. But this relies on perfect
alignment onto the SLM of the pulse shaper. Any slight alignment deviations
that occur in time, which are common for many laser systems, will result in a
considerable change in the pump/Stokes excitation bandwidth (Figure 12). If
the excitation bandwidth changes between the measurement of a CARS spectrum
and its corresponding NR Ar spectrum, then the division by the NR spectrum to
correct for the pump/Stokes excitation bandwidth will result in an incorrect
best fit temperature, leading to best fit temperature outliers. This
highlights the need for taking many NR Ar spectra throughout an experimental
run when using a pulse shaper, so that the pump/Stokes excitation bandwidth is
the same for a CARS spectrum and its corresponding NR argon spectrum.

Pump/Stokes beam alignment drift should affect low and high pressure spectra
similarly. But from the full set of results (Figure 17), it can be seen that
most of the less accurate best fit temperatures occur at higher pressures. One
explanation for this is that while the pump/Stokes excitation differences due
to alignment drift affect all spectra equally, the NR spectra start to narrow
with increasing pressure, as discussed before (Figure 12). There could be a
variation in the degree of narrowing depending on the initial pump/Stokes
excitation bandwidth at atmospheric pressure. This could lead to a compounding
of the two effects (different pump/Stokes excitation bandwidth between the N2
CARS and NR Ar spectrum and the narrowing of the excitation bandwidth with
increasing pressure), leading to more inaccurate high pressure spectra than
low pressure spectra, while maintaining similar average accuracies, as the
effect is random. This is further amplified by high pressure spectra, in
general, having a lower signal-to-noise ratio, resulting in less accurate
fits. This decrease in signal strength at high pressures is due to the time
domain molecular response only increasing linearly with number density, but
decreasing exponentially with pressure, leading to an overall decrease in the
signal (see Figure 4). Unfortunately, it is not possible to use a pulse shaper
compensation mask at high pressure, as there is no way to get high pressure
feedback signal to the pulse shaper algorithm. A potential solution for a
chirp precompensation of the pump/Stokes pulse that is more stable with time
could be the use of chirped mirrors. These mirrors induce a larger group delay
dispersion to longer wavelengths than to shorter wavelengths. This
precompensates a beam for the chirp induced while passing through an optical
system, for example fused silica windows. However, chirped mirrors only
correct for linear chirp, while the pulse shaper corrects for non-linear chirp
as well.

Another factor that could contribute to the lower accuracy at higher pressures
is uncertainty in the experimentally measured linewidths that are used in the
model. Small errors in the linewidths have an increasingly larger effect on
the model spectrum (and hence the best fit temperature), with increasing
pressure, due to the exponential dependence of the CARS signal on the
linewidths (see Equation 5).

##  6 Conclusion

In conclusion, we have developed a hybrid rotational fs/ps CARS experimental
setup that allows the collection of clean and well resolved spectra over a
wide range of temperatures and pressures, in previously unexplored T-P
regimes. This has been done using a pulse shaper to create a near TL
pump/Stokes pulse at the interaction volume and by making a 5.5 ps probe beam
in an SHBC setup that allows probing of the molecular response close to but
not overlapping the nonresonant response. Further, we have developed a
computational model that generates theoretical CARS spectra and can be used
for fitting the experimental spectra to infer temperature from the spectral
structure. While being based on a few assumptions and simplifications, much of
the complexity of the CARS theory is retained in the model. The temperature
fits reveal that our model is relatively accurate with 84% of all acquired
experimental spectra fitting with an error of 10% or less. Even though more
accurate at lower pressures, the model still gives good fits at high
pressures, both qualitatively and quantitatively. However, it was also found
that there are some less accurate fits. This should be taken into
consideration for future experiments.

Now that we have shown that accurate thermometry is possible with HR-CARS at
the high temperatures and pressures encountered in real combustion
environments, the next step will be to further develop and test this technique
to achieve even higher accuracy of temperature measurements. One day, this
experiment could be performed in a real optical engine, to shed light on the
robustness and suitability of this technique for temperature measurements in
turbulent systems.

This material is based upon the work supported by an award from the United
States Department of Energy’s Office of Science Early Career Research Program,
which supported the development of the CARS model, the laser equipment, and
C.J.K. The construction of the high pressure cell flow reactor, the
femtosecond pulse shaper, and T.L.C. was supported by the Laboratory Directed
Research and Development (LDRD) program at Sandia National Laboratories, which
is a multimission laboratory managed and operated by National Technology and
Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell
International, Inc., for the U.S. Department of Energy’s National Nuclear
Security Administration under Contract No. DE-NA0003525. The views expressed
in this article do not necessarily represent the views of the U.S. Department
of Energy or the United States Government.

B. Peterson gratefully acknowledges financial support from the European
Research Council (ERC grant #759546).

Disclosures. The authors declare no conflicts of interest.

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# A NOTE ON IDEMPOTENT SEMIRINGS

George Janelidze  Department of Mathematics and Applied Mathematics,
University of Cape Town, Rondebosch 7700, South Africa  Dedicated to Themba
Dube on the occasion of his 65th birthday<EMAIL_ADDRESS>and
Manuela Sobral  CMUC and Departamento de Matemática, Universidade de Coimbra,
3001–501 Coimbra, Portugal<EMAIL_ADDRESS>
###### Abstract.

For a commutative semiring $S$, by an $S$-algebra we mean a commutative
semiring $A$ equipped with a homomorphism $S\to A$. We show that the
subvariety of $S$-algebras determined by the identities $1+2x=1$ and $x^{2}=x$
is closed under non-empty colimits. The (known) closedness of the category of
Boolean rings and of the category of distributive lattices under non-empty
colimits in the category of commutative semirings both follow from this
general statement.

###### Key words and phrases:

commutative semiring, non-empty colimit, coreflective subcategory, Boolean
algebra, distributive lattice

###### 2010 Mathematics Subject Classification:

16Y60, 18A30, 18A40, 06E20, 06E75, 06D75

Partially supported by the Centre for Mathematics of the University of Coimbra
– UID/MAT/00324/2020

##  1\. Introduction

Let us recall:

1.1. A _commutative semiring_ is an algebraic structure of the form
$S=(S,0,+,1,\cdot)$ in which $(S,0,+)$ and $(S,1,\cdot)$ are commutative
monoids with

$x0=0,\,\,\,x(y+z)=xy+xz$

for all $x,y,z\in S$. Here and below we use standard notational agreements:
e.g. $xy+xz$ means $(x\cdot y)+(x\cdot z)$. The category of commutative
semirings will be denoted by $\mathsf{CSR}$; as expected, its morphisms are
semiring homomorphisms, that is, maps $f:S\to S^{\prime}$ of (commutative)
semirings with

$f(0)=0,\,\,\,f(s+t)=f(s)+f(t),\,\,\,f(1)=1,\,\,\,f(st)=f(s)f(t)$

for all $s,t\in S$.

1.2. For a (commutative) semiring $S$, an $S$-_module_ (or $S$-_semimodule_)
is an algebraic structure of the form $A=(A,0,+,h)$ in which $(A,0,+)$ is a
commutative monoid and $h:S\times A\to A$ is a map written as $(s,a)\mapsto
sa$ that has

$1a=a,\,\,\,s(ta)=(st)a,\,\,\,s0=0,\,\,\,s(a+b)=sa+sb,\,\,\,0a=0,\,\,\,(s+t)a=sa+ta$

for all $s,t\in S$ and $a,b\in A$. The category of $S$-modules will be denoted
by $S$-$\mathsf{mod}$.

1.3. For $S$ above, a _commutative $S$-algebra_ is a pair $(A,f)$ in which $A$
is a commutative semiring, and $f:S\to A$ is a semiring homomorphism.
Accordingly, the category of commutative $S$-algebras is just the comma
category $(S\downarrow\mathsf{CSR})$. Equivalently, a commutative $S$-algebra
can be defined as an algebraic structure of the form $A=(A,0,+,h,1,\cdot)$ in
which $A=(A,0,+,h)$ is an $S$-module and $A=(A,0,+,1,\cdot)$ is a commutative
semiring with $s(ab)=(sa)b$ for all $s\in S$ and $a,b\in A$; we will then have
$sa=f(s)a$.

1.4. If $S$ is a commutative ring, then $(S\downarrow\mathsf{CSR})$ is the
ordinary category of commutative (unital) $S$-algebras. In particular: if
$S=\mathbb{Z}$ is the ring of integers, then $(S\downarrow\mathsf{CSR})$ is
the category $\mathsf{CRings}$ of commutative rings; if $S=\\{0,1\\}$ with
$1+1=0$, then $(S\downarrow\mathsf{CSR})$ is the category
$\mathsf{CRings}_{2}$ of commutative rings of characteristic $2$ (=the
category of commutative algebras over the two-element field). The category
$\mathsf{CRings}_{2}$ contains the category $\mathsf{BRings}$ of Boolean rings
(=commutative rings satisfying the identity $x^{2}=x$).

1.5. If $S=\\{0,1\\}$ with $1+1=1$, then $(S\downarrow\mathsf{CSR})$ is the
category $\mathsf{AICSR}$ of additively idempotent commutative semirings
(=commutative semirings satisfying the identity $2=1$, or, equivalently, the
identity $2x=x$). This category contains the category $\mathsf{DLat}$ of
distributive lattices.

In this paper we present a general result (Corollary 2.3 in the next section)
on $S$-semialgebras, which implies (known) closedness of the categories of
Boolean rings and of distributive lattices under non-empty colimits (or,
equivalently, just under binary coproducts) in the category of commutative
semirings.

##  2\. The general case

Let us first mention an obvious general fact:

######  Lemma 2.1.

Let $\mathcal{V}$ be a variety of universal algebras, $\mathcal{W}$ a
subvariety of $\mathcal{V}$, and $I$ the initial object of $\mathcal{W}$ (=the
free algebra in $\mathcal{W}$ on the empty set). If
$\mathcal{W}\approx(I\downarrow\mathcal{W})$ is coreflective in
$(I\downarrow\mathcal{V})$, then $\mathcal{W}$ is closed under non-empty
colimits in $\mathcal{V}$.∎

Then we take:

  * •

$\mathcal{V}=(S\downarrow\mathsf{CSR})$;

  * •

$\mathcal{W}=(S\downarrow\mathsf{CSR})^{*}$ to be the subvariety of
$(S\downarrow\mathsf{CSR})$ consisting of all $S$-algebras satisfying the
identities $1+2x=1$ and $x^{2}=x$. This makes $I\approx S/E$, where $E$ is the
smallest congruence on $S$ containing $(1+2s,1)$ and $(s^{2},s)$ for each
$s\in S$.

######  Theorem 2.2.

Let $I\approx S/E$ be as above. The variety

$(S\downarrow\mathsf{CSR})^{*}\approx(I\downarrow(S\downarrow\mathsf{CSR})^{*})$

of commutative $S$-algebras satisfying the identities $1+2x=1$ and $x^{2}=x$
is a coreflective subcategory of
$(I\downarrow\mathsf{CSR})\approx(I\downarrow(S\downarrow\mathsf{CSR}))$.

###### Proof.

It suffices to show that, for each $A\in(I\downarrow\mathcal{V})$, the set

$A^{\prime}=\\{a\in A\mid 1+2a=1\,\,\&\,\,a^{2}=a\\}$

is a subalgebra of $A$, that is, to show the following:

  * (i)

$a,b\in A^{\prime}\Rightarrow a+b\in A^{\prime}$;

  * (ii)

for each $s\in S$, $a\in A^{\prime}\Rightarrow sa\in A^{\prime}$;

  * (iii)

$1\in A^{\prime}$;

  * (iv)

$a,b\in A^{\prime}\Rightarrow ab\in A^{\prime}$;

Indeed, (i): Suppose $a$ and $b$ are in $A^{\prime}$. Then
$1+2(a+b)=1+2a+2b=1+2b=1$ and
$(a+b)^{2}=a^{2}+2ab+b^{2}=a+2ab+b=a(1+2b)+b=a+b$.

(iii): $1+2\cdot 1=1$ since this equality holds in $I$.

(iv): Suppose $a$ and $b$ are in $A^{\prime}$. Then

$1+2ab=1+2a+2ab=1+a+a(1+2b)=1+a+a=1+2a=1$

and $(ab)^{2}=a^{2}b^{2}=ab$.

(ii) follows from (iv) since $sa=(s1)a$ and $(s1)$ is in $A^{\prime}$ (since
$s1$ is the image of class of $s$ under the homomorphism $I\to A$). ∎

From Lemma 3.1 and Theorem 3.2 we obtain:

######  Corollary 2.3.

The variety $(S\downarrow\mathsf{CSR})^{*}$ of commutative $S$-algebras
satisfying the identities $1+2x=1$ and $x^{2}=x$ is closed under non-empty
colimits in the variety $(S\downarrow\mathsf{CSR})$ of all commutative
$S$-algebras.∎

Taking $S$ to be the ring of natural numbers, we obtain the following special
cases of Theorem 2.2 and Corollary 2.3:

######  Corollary 2.4.

The variety $\mathsf{CSR}^{*}$ of commutative semirings satisfying the
identities $1+2x=1$ and $x^{2}=x$ is coreflective in the variety
$(\\{0,1,2\\}\downarrow\mathsf{CSR})$, where $1+2=1$ in $\\{0,1,2\\}$.∎

######  Corollary 2.5.

The variety $\mathsf{CSR}^{*}$ above is closed under non-empty colimits in
$\mathsf{CSR}$.∎

##  3\. Boolean rings and distributive lattices

If an object $A$ of $(\\{0,1,2\\}\downarrow\mathsf{CSR})$ belongs to
$(\\{0,1\\}\downarrow\mathsf{CSR})$ with $1+1=0$ in $\\{0,1\\}$ making
$(\\{0,1\\}\downarrow\mathsf{CSR})=\mathsf{CRings}_{2}$, then

$\\{a\in A\mid 1+2a=1\,\,\&\,\,a^{2}=a\\}=\\{a\in A\mid
2a=0\,\,\&\,\,a^{2}=a\\}.$

But if it is the case with $1+1=1$ making
$(\\{0,1\\}\downarrow\mathsf{CSR})=\mathsf{AICSR}$, then

$\\{a\in A\mid 1+2a=1\,\,\&\,\,a^{2}=a\\}=\\{a\in A\mid
1+a=1\,\,\&\,\,a^{2}=a\\}.$

Therefore we obtain the commutative diagram

$\textstyle{\mathsf{CRings}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\\{0,1,2\\}\downarrow\mathsf{CSR})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathsf{AICSR}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathsf{BRings}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathsf{CSR}^{*}}$$\textstyle{\mathsf{DLat}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

where the horizontal arrows are inclusion functors while the left-hand and
right-hand vertical arrows are the coreflections induced by the coreflection
of Corollary 2.4 represented by the middle vertical arrow. Since
$\mathsf{CRings}_{2}$ and $\mathsf{AICSR}$ both being of the form
$(\\{0,1\\}\downarrow\mathsf{CSR})$ (with different $1+1$ in $\\{0,1\\}$) are
closed in $\mathsf{CSR}$ under non-empty colimits, we conclude that both
$\mathsf{BRings}$ and $\mathsf{DLat}$ are also closed in $\mathsf{CSR}$ under
non-empty colimits. That is, as promised in our Introduction, these two known
results follow from what we have done in general (in Section 2).

##  4\. Two additional remarks

4.1. The Reader might ask, what is special about $(S\downarrow\mathsf{CSR})$?
The answer consists of the following observations:

  * •

$(S\downarrow\mathsf{CSR})$ is the category of commutative monoids in the
monoidal category $S$-$\mathsf{mod}$ having therefore ‘good’ colimits; indeed,
its binary coproducts are given by tensor products.

  * •

The monoidal category structure of $S$-$\mathsf{mod}$ is determined by the
fact that it is a commutative variety of universal algebras.

  * •

A commutative variety of universal algebras is semi-additive if and only if it
is of the form $S$-$\mathsf{mod}$ for some commutative semiring $S$. This
immediately follows from the equivalence $1.\\!\Leftrightarrow 5.$ in Theorem
2.1 of [2], which refers to [1] for the proof.

4.2. The coreflectivity of $\mathsf{DLat}$ in $\mathsf{AICSR}$ is a ‘finitary
copy’ of the coreflectivity of the category of frames in the category of
quantales, see Section C1.1 of [3]: in fact $A_{f}$ on Page 479 there is the
same as our $\\{a\in A\mid 1+a=1\,\,\&\,\,a^{2}=a\\}$.

## References

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