# Algorithms for Least-Squares Noncartesian MR Image Reconstruction

Tobias C Wood
<EMAIL_ADDRESS>Department of Neuroimaging, King’s College London

## Abstract

Iterative least-squares MR reconstructions typically use the Conjugate
Gradient algorithm, despite known numerical issues. This paper demonstrates
that the more recent LSMR algorithm has favourable numerical properties, and
is to be preferred in situations where Toeplitz embedding cannot be used to
accelerate the Conjugate Gradient method.

## Introduction

The seminal paper of Pruessman et al[1] introduced the concept of formulating
MR image reconstruction as a least squares problem:

$Ax=b+\mathnormal{\epsilon}$ (1)

where $\bf x$ represents the desired image as a vector of complex-valued voxel
values, $\bf b$ is a similar vector containing the acquired k-space samples,
$\bf\epsilon$ is complex-valued Gaussian-distributed noise, and $\bf A$ is the
system matrix (often also denoted as the encoding operator/matrix $\bf E$ in
the MR literature). This represents all the steps involved in the MR image
formation process, which generally include splitting the image into multiple
receiver channels, Fourier transformation, and sampling along the desired
trajectory.

In general for MRI the matrix $\bf A$ is too large to store in memory or
efficiently invert, which precludes using the closed-form solution to (1).
Instead Pruessman et al. expressed the sensitivity, Fourier, and sampling
steps as linear operators and solved (1) using the iterative Conjugate
Gradient (CG) algorithm [1]. This has become popular enough that the term
“cgSENSE” has entered the MRI lexicon [2]. CG is incredibly simple to
implement, features superior convergence performance compared to simple
gradient descent and Toeplitz embedding can be used to avoid repeated gridding
operations which leads to a significant computation speed increase for
noncartesian trajectories [3, 4, 5].

However, CG has some numerical properties that are not well suited to MR
reconstruction [1]. These problems arise because, in general, the system
matrix $\bf A$ for MRI is not square or symmetric (due to any or all of
multiple channels, undersampling, and a noncartesian trajectory). CG requires
a positive definite system matrix, and to guarantee this instead of directly
solving (1) we solve the normal equations

$A^{\dagger}Ax=A^{\dagger}b$ (2)

where † denotes the adjoint (conjugate transpose). This squares the condition
number of the system, which makes it more sensitive to random noise and leads
to very slow convergence of 3D noncartesian trajectories (which have a large
condition number due to the repeated sampling of the center of k-space). In
addition preconditioning, a standard method to reduce the effective condition
number, can only be used in image space and not k-space [6]. Non-iterative
noncartesian reconstructions use Sample Density Compensation (SDC) weightings
in k-space to reconstruct meaningful images [7, 8, 9]. While SDC weightings
can be, and often are, inserted in an ad-hoc way into cgSENSE, this results in
noise inflation [1, 5].

My particular motivation is reconstructing Zero Echo-Time (ZTE) images, which
necessitate a 3D radial center-out trajectory for which the above issues are
particularly acute [10]. ZTE can capture signals from protons with short
transverse relaxation times which are normally MR invisible [11, 12, 13]. A
standard clinical MRI setup contains foam cushioning and structural plastic in
the receive coil that contain significant amounts of such protons, and so
contain signals outside the nominal field of view. This enforces an enlarged
reconstruction field of view to avoid aliasing. Toeplitz embedding would
require an additional eight-fold memory requirement (two times in each
dimension) [5]. For reasonably sized images, for example 1$\bf\mathrm{mm}$
isotropic or better full brain images, the large memory requirements this
imposes quickly become impractical. This limitation motivated a search for
alternative algorithms.

Two such algorithms are LSQR [14] and the much more recent LSMR [15]111The
original papers for LSQR and LSMR do not define the names as acronyms. However
I believe they stand for “Least-Squares via QR factorization” and “Least-
Squares Minimum Residual” respectively. These both use a Golub-Kahan bi-
diagonolization process, which is computationally similar to CG. To quote the
Stanford University Systems Optimization Laboratory website: “[LSQR] is
algebraically equivalent to applying CG to the normal equation but has better
numerical properties, especially if $\bf A$ is ill-conditioned”, and “LSQR
reduces $\bf|r|$ monotonically (where $\bf r=b-Ax$ … ) On least-squares
problems, if an approximate solution is acceptable (stopping tolerances quite
large), LSMR may be a preferable solver because it reduces both $\bf|r|$ and
$\bf|A^{\prime}r|$ monotonically and may be able to terminate significantly
earlier”. Because both algorithms conduct operations in image space and
k-space they support preconditioning in either domain [16]. Ong et al recently
derived a k-space preconditioner that gave superior results to SDC in the
context of regularized reconstructions using the Primal-Dual Hybrid Gradient
(PDHG) algorithm [6]. Both algorithms support simple Tikhonov regularization
through a simple algorithmic alteration.

In this paper I demonstrate that, when Toeplitz embedding is impractical, LSQR
gives comparable performance to CG while LSMR outperforms both. LSMR hence may
be of interest to the wider MR community for reconstructing large noncartesian
MR images.

## Methods

I compared the performance of the CG, LSQR and LSMR algorithms using two
different datasets: the 2D radial dataset from the 2020 ISMRM Reproducibility
Challenge [2] and an in-vivo 3D radial ZTE scan. Acquisition details for the
2D dataset are given in the reference. The 3D dataset was acquired on a 3T
scanner equipped with a 48-channel head coil (GE Healthcare) at
1$\bf\mathrm{mm}$ isotropic resolution, 220 isotropic matrix size, flip-angle
$\bf\mathrm{2$\bf\mathrm{\SIUnitSymbolDegree}$}$,
$\bf\mathrm{TR/TI/T_{recovery}=2/800/600$\bf\mathrm{ms}$}$, 2x radial
oversampling, $\bf\mathrm{2\pi}$ spoke undersampling, 512 spokes per segment
and a phyllotaxis trajectory [17].

All reconstructions were carried out with the RIESLING toolbox [18]. The
Jupyter notebooks used for the analysis are available at
https://github.com/spinicst/riesling-algorithms. Reconstructions used Apple
MacBook Pro with an M1 Pro chip and 32 gigabytes of RAM. The 3D data was first
compressed from to 12 virtual channels [19], and then the same steps were
followed for both 2D and 3D. Sensitivity maps were extracted directly from the
fully-sampled central k-space region [20]. For CG, SDC weights were
calculating with Pipe’s method [9] while for LSQR and LSMR I used Ong’s
single-channel k-space preconditioner [6]. Both datasets were then
reconstructed twice with each algorithm, once without SDC/preconditioning as
appropriate and once without, yielding six different reconstructions for each
dataset. All algorithms were terminated at a fixed limit of 32 iterations. The
results were compared both at this final iteration, and earlier at a somewhat
arbitrary choice of the sixth iteration for the 2D dataset and the 4th
iteration for 3D.

## Results

Figure 1: The logarithm of the norm of the image-space residual for the 2D
(left) and 3D (right) datasets for all the algorithms investigated.

Figure 2: Reconstructed images at the final iteration for the 2D data without
(left) and with (right) SDC or preconditioning. The images are shown on the
diagonal and percent (normalised to the highest intensity value in the CG
image) differences between them on the upper off-diagonals.

Figure 3: Reconstructed images at the final iteration for the 3D data without
(left) and with (right) SDC or preconditioning. The images are shown on the
diagonal and percent (normalised to the highest intensity value in the CG
image) differences between them on the upper off-diagonals.

Figure 1 shows the convergence behaviour for all algorithms and both datasets.
The quantity of interest is the norm of the image-space residual, which is the
$\bf|A^{\dagger}r|$ term for LSQR and LSMR as the primary residual is in
k-space. CG and LSQR produce near identical results without
SDC/preconditioing, including oscillatory behaviour with iterations where the
residual temporarily increases. The addition of SDC/preconditioning reduces
the residual norm for all three algorithms, and leads to differences in the
residual between CG and LSQR. LSMR produces a smaller final residual in all
cases after a few iterations, although in the 2D case CG does outperform it
initially. The computation of all three algorithms was approximately the same,
at 7 iterations per second for 2D and 1 iteration per 30 seconds for 3D
(including the time taken to save per-iteration data to generate the figures
in this paper).

Figures 2&3 show the reconstructed images at the final iteration for each
algorithm and the differences between them. Matching the residual results,
without SDC/preconditioning, CG and LSQR produce essentially identical images.
The 2D reconstructions are crisp and high quality, while the 3D
reconstructions are blurred. Significant signal can be observed outside the
head from cushioning in the 3D images due to the ZTE acquisition. The images
from LSMR are nearly indistinguishable to CG and LSQR, but in the 2D case
small but elevated differences are evident on the edges of the skull while for
the 3D data much larger differences are apparent, including within the brain
on grey matter/white matter boundaries.

The addition of SDC/preconditioning leads to differences between CG and LSQR,
which perhaps show some structure in the 2D case but are only noise in the 3D
case. There are no anatomical features in the difference between LSMR and the
other two algorithms, but LSMR arguably displays a lower background noise
level. The 3D images with SDC/preconditioning appear very noisy.

Figure 4: Reconstructed images at iteration 6 for the 2D data without (left)
and with (right) SDC or preconditioning. The bottom row shows the difference
from the previous iteration.

Figure 5: Reconstructed images iteration 4 for the 3D data without (left) and
with (right) SDC or preconditioning. The bottom row shows the difference from
the previous iteration.

Figures 5&6 show the reconstructed images when terminated early, along with
the difference from the previous iteration. In the 2D case, without
SDC/preconditioning the algorithm has clearly not finished, with significant
anatomy present in the change from the previous iteration. With
SDC/preconditioning anatomical features are largely absent from the update
image but for both CG and LSQR low spatial-frequency changes are obvious. In
contrast, with the exception of the edges of the skull, LSMR shows a much
flatter update image.

In 3D this pattern is repeated, but more pronounced with the images lacking
SDC/preconditioning being exceedingly blurry. With SDC/preconditioning, the
update from the previous iteration is again flattest for LSMR, and overall the
image quality is significantly better than when the algorithm was run for 32
iterations.

## Discussion

The results above demonstrate the superior numerical performance of the LSMR
algorithm for noncartesian MR reconstruction. LSMR produced the lowest
residual values, but more importantly demonstrates monotonic convergence
behaviour that is safe to terminate earlier than either CG or LSQR. When
preconditioning is used, the image quality after only a handful of iterations
is equivalent (2D) or superior (3D) to that without preconditioning after many
iterations. There is essentially no computational drawback to using LSMR
compared to CG. The majority of algorithmic time is spent in the forward and
adjoint Non-Uniform Fast Fourier Transform (NUFFT) operator and is the same
for both (when Toeplitz embedding is not used). LSMR does require a larger
number of subsequent scalar calculations to find the update step-size, and is
hence more tedious to implement, but these calculations take negligible run
time.

The question of whether a Toeplitz-embedded reconstruction using CG would
still be advantageous over LSMR is one I do not attempt to answer here. Such a
comparison would involve numerous trade-offs such as memory consumption versus
run-time performance and reconstruction accuracy that will depend on the
specifics of the particular MR reconstruction problem. I included the 2D data,
which could easily be used with Toeplitz embedding, in this paper to provide
an easy reference point to the existing literature, but my particular interest
is the 3D ZTE data where for reasons outlined above Toeplitz embedding is not
always possible.

I found a limited number of references to using LSQR in the MRI literature
[21, 22, 23]. Of note, Hoge et al exploited the way Tikhonov regularization
works in LSQR to provide automatic tuning of the regularization parameter
[21]. LSMR has previously been used to improve image quality for Cartesian
GRAPPA [24] and was recently used as the comparison method against a Total-
Variation regularized reconstruction of PROPELLER data [25].

I am not aware of an investigation of the algorithms for general noncartesian
imaging as presented here. I hypothesise this is likely due to the fact that
an efficient preconditioner for noncartesian reconstruction, which is
essential for both efficiency and image quality, was only discovered recently
[6]. The single-channel preconditioner as used here can be precomputed for a
particular trajectory, as can SDC weightings, and during each iteration only
requires a single element-wise multiplication operation in k-space which takes
minimal time.

While slightly beyond the scope of this paper, it is worth mentioning that
LSMR (or LSQR) can be used as the inner-solver for the Alternating Directions
Method-of-Multipliers (ADMM) algorithm which has become popular for MRI
reconstruction [26, 27, 28]. An example of how to formulate this with LSQR can
be found on the website associated with Boyd et al [29].

## Conclusion

In situations where Toeplitz embedding is impractical the LSMR algorithm has
numerical benefits over the Conjugate Gradient algorithm and negligible
drawbacks.

## Acknowledgements

I thank Emil Ljungberg for his encouragement and reading a draft of this
manuscript.

## References

  * [1] Klaas P. Pruessmann, Markus Weiger, Peter Börnert, and Peter Boesiger.  Advances in sensitivity encoding with arbitrary k-space trajectories.  Magnetic Resonance in Medicine, 46(4):638–651, October 2001.
  * [2] Oliver Maier, Steven Hubert Baete, Alexander Fyrdahl, Kerstin Hammernik, Seb Harrevelt, Lars Kasper, Agah Karakuzu, Michael Loecher, Franz Patzig, Ye Tian, Ke Wang, Daniel Gallichan, Martin Uecker, and Florian Knoll.  CG‐SENSE revisited: Results from the first ISMRM reproducibility challenge.  Magnetic Resonance in Medicine, November 2020.
  * [3] FTAW Wajer and KP Pruessmann.  Major speedup of reconstruction for sensitivity encoding with arbitrary trajectories.  In Proc. Intl. Soc. Mag. Res. Med, page 767, 2001.
  * [4] J.A. Fessler, Sangwoo Lee, V.T. Olafsson, H.R. Shi, and D.C. Noll.  Toeplitz-based iterative image reconstruction for MRI with correction for magnetic field inhomogeneity.  IEEE Transactions on Signal Processing, 53(9):3393–3402, September 2005.
  * [5] Corey A. Baron, Nicholas Dwork, John M. Pauly, and Dwight G. Nishimura.  Rapid compressed sensing reconstruction of 3D non-Cartesian MRI.  Magnetic Resonance in Medicine, 79(5):2685–2692, May 2018.
  * [6] Frank Ong, Martin Uecker, and Michael Lustig.  Accelerating Non-Cartesian MRI Reconstruction Convergence Using k-Space Preconditioning.  IEEE Transactions on Medical Imaging, 39(5):1646–1654, May 2020\.
  * [7] James G. Pipe.  Reconstructing MR images from undersampled data: Data-weighting considerations.  Magnetic Resonance in Medicine, 43(6):867–875, June 2000.
  * [8] Kenneth O. Johnson and James G. Pipe.  Convolution kernel design and efficient algorithm for sampling density correction.  Magnetic Resonance in Medicine, 61(2):439–447, February 2009.
  * [9] Nicholas R. Zwart, Kenneth O. Johnson, and James G. Pipe.  Efficient sample density estimation by combining gridding and an optimized kernel: Efficient Sample Density Estimation.  Magnetic Resonance in Medicine, 67(3):701–710, March 2012.
  * [10] Emil Ljungberg, Nikou Damestani, Tobias C Wood, David J Lythgoe, Fernando Zelaya, Steven C R Williams, Ana Beatriz Solana, Gareth J Barker, and Florian Wiesinger.  Silent zero TE MR neuroimaging: Current state-of-the-art and future directions.  Progress in Nuclear Magnetic Resonance Spectroscopy, page 21, 2021\.
  * [11] Markus Weiger and Klaas P. Pruessmann.  Short-T2 MRI: Principles and recent advances.  Progress in Nuclear Magnetic Resonance Spectroscopy, 114-115:237–270, October 2019.
  * [12] Markus Weiger, Romain Froidevaux, Emily Louise Baadsvik, David Otto Brunner, Manuela Barbara Rösler, and Klaas Paul Pruessmann.  Advances in MRI of the myelin bilayer.  NeuroImage, 217:116888, August 2020.
  * [13] Emily Louise Baadsvik, Markus Weiger, Romain Froidevaux, Manuela Barbara Rösler, David Otto Brunner, Lena Öhrström, Frank Jakobus Rühli, Patrick Eppenberger, and Klaas Paul Pruessmann.  High‐resolution MRI of mummified tissues using advanced short‐T $\bf{}_{\textrm{2}}$ methodology and hardware.  Magnetic Resonance in Medicine, 85(3):1481–1492, March 2021.
  * [14] Christopher C. Paige and Michael A. Saunders.  LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares.  ACM Transactions on Mathematical Software, 8(1):43–71, March 1982\.
  * [15] David Chin-Lung Fong and Michael Saunders.  LSMR: An Iterative Algorithm for Sparse Least-Squares Problems.  SIAM Journal on Scientific Computing, 33(5):2950–2971, January 2011\.
  * [16] A. Montoison, D. Orban, and contributors.  Krylov.jl: A Julia basket of hand-picked Krylov methods, June 2020\.
  * [17] Emil Ljungberg, Tobias C. Wood, Ana Beatriz Solana, Steven C. R. Williams, Gareth J. Barker, and Florian Wiesinger.  Motion corrected silent ZTE neuroimaging.  Magnetic Resonance in Medicine, page mrm.29201, April 2022.
  * [18] Tobias Wood, Emil Ljungberg, and Florian Wiesinger.  Radial Interstices Enable Speedy Low-volume Imaging.  Journal of Open Source Software, 6(66):3500, October 2021.
  * [19] Feng Huang, Sathya Vijayakumar, Yu Li, Sarah Hertel, and George R. Duensing.  A software channel compression technique for faster reconstruction with many channels.  Magnetic Resonance Imaging, 26(1):133–141, January 2008.
  * [20] Ernest N. Yeh, Matthias Stuber, Charles A. McKenzie, Rene M. Botnar, Tim Leiner, Michael A. Ohliger, Aaron K. Grant, Jacob D. Willig-Onwuachi, and Daniel K. Sodickson.  Inherently self-calibrating non-cartesian parallel imaging.  Magnetic Resonance in Medicine, 54(1):1–8, July 2005.
  * [21] W.S. Hoge, M.E. Kilmer, S.J. Haker, D.H. Brooks, and W.E. Kyriakos.  Fast Regularized Reconstruction of Non-Uniformly Subsampled Parallel MRI Data.  In 3rd IEEE International Symposium on Biomedical Imaging: Macro to Nano, 2006., pages 714–717, Arlington, Virginia, USA, 2006. IEEE.
  * [22] Peng Qu, Jing Luo, Bida Zhang, Jianmin Wang, and Gary X. Shen.  An improved iterative SENSE reconstruction method.  Concepts in Magnetic Resonance Part B: Magnetic Resonance Engineering, 31B(1):44–50, February 2007.
  * [23] Justin P. Haldar.  Autocalibrated loraks for fast constrained MRI reconstruction.  In 2015 IEEE 12th International Symposium on Biomedical Imaging (ISBI), pages 910–913, Brooklyn, NY, USA, April 2015. IEEE.
  * [24] Daniel S. Weller, Jonathan R. Polimeni, Leo Grady, Lawrence L. Wald, Elfar Adalsteinsson, and Vivek K. Goyal.  Sparsity-Promoting Calibration for GRAPPA Accelerated Parallel MRI Reconstruction.  IEEE Transactions on Medical Imaging, 32(7):1325–1335, July 2013\.
  * [25] Hsin-Chia Chen, Haw-Chiao Yang, Chih-Ching Chen, Seb Harrevelt, Yu-Chieh Chao, Jyh-Miin Lin, Wei-Hsuan Yu, Hing-Chiu Chang, Chin-Kuo Chang, and Feng-Nan Hwang.  Improved Image Quality for Static BLADE Magnetic Resonance Imaging Using the Total-Variation Regularized Least Absolute Deviation Solver.  Tomography, 7(4):555–572, October 2021.
  * [26] Martin Uecker, Frank Ong, Jonathan I Tamir, Dara Bahri, Patrick Virtue, Joseph Y Cheng, Tao Zhang, and Michael Lustig.  Berkeley advanced reconstruction toolbox.  In Proc. Intl. Soc. Mag. Reson. Med, volume 23, 2015.
  * [27] Jonathan I. Tamir, Martin Uecker, Weitian Chen, Peng Lai, Marcus T. Alley, Shreyas S. Vasanawala, and Michael Lustig.  T2 shuffling: Sharp, multicontrast, volumetric fast spin‐echo imaging.  Magnetic Resonance in Medicine, 77(1):180–195, January 2017.
  * [28] Jakob Assländer, Martijn A. Cloos, Florian Knoll, Daniel K. Sodickson, Jürgen Hennig, and Riccardo Lattanzi.  Low rank alternating direction method of multipliers reconstruction for MR fingerprinting: Low Rank ADMM Reconstruction.  Magnetic Resonance in Medicine, 79(1):83–96, January 2018.
  * [29] Stephen Boyd.  Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers.  Foundations and Trends® in Machine Learning, 3(1):1–122, 2010\.


# Diode laser overtone spectroscopy of methane at 780 nm

A. Lucchesini<EMAIL_ADDRESS>Istituto Nazionale di Ottica - CNR - S.S.
“Adriano Gozzini”
Area della Ricerca - Pisa - Italy

###### Abstract

A tunable diode laser spectrometer has been employed to examine the overtone
absorption lines of 12CH4 in the region around 12780 cm-1 (780 nm) at room
temperature. The spectrometer sources are commercially available tunable diode
lasers operating in “free-running” mode. The measurement of the line positions
within 0.01 cm-1 has been possible by the use of the wavelength modulation
spectroscopy and the second harmonic detection techniques that permitted the
observation of minimum optical absorbances of the order of $\approx 10^{-6}$.
The weakest observed lines of the band have absorption cross sections of the
order of $1\times 10^{-25}$ cm2/molecule, that is $\simeq 0.3$ km-1/amagat.
For some of them self-, air-, He- and H2-broadening coefficients have been
obtained.

##  I Introduction

12CH4 is an extremely interesting molecule present in the spectra of the outer
planets of the solar system like Uranus, Neptune, Jupiter, Saturn and the
relative satellites, in particular Titan [1], where it is the second most
abundant gas in the atmosphere; both methane and ethane have also been found
in the Kuiper Belt [2]. In spite of their weakness, overtone and combination
tone absorption lines of the methane gas are observable in the absorbing
layers of the planets [3] because these are much thicker than the ones
obtainable in the laboratory. The knowledge of the methane optical resonances
and their behavior with pressure is important for a better knowledge of the
planetary atmospheres. Conventional spectroscopy measurements of this
spherical top molecule in the visible and in the near infrared have been
systematically performed in the past by analyzing the telescope photographic
plates [4] or by using long path White cells in the laboratories [5, 6], but
to detect the weak overtone bands, advanced technologies like the intracavity
laser spectroscopy (ILS) have been used [7, 8].
After the previous work on the $\nu_{1}+3\nu_{3}$ methane absorption band [9]
another spectroscopic work based on the use of the semiconductor diode lasers
(DLs) is presented here. The frequency modulation (FM) technique comes to help
when dealing with very weak absorption resonances taking advantage of one of
the more interesting properties of the DLs, that is the modulability through
the injection current. In this work we apply the FM spectroscopy, usually
called “wavelength modulation spectroscopy” (WMS) when the value of the
frequency is chosen much lower than the resonance line-width, and the second
harmonic detection techniques to the methane absorptions around 780 nm, where
the very weak CH4 combination overtone band $3\nu_{1}+\nu_{3}+(\nu_{2}$ or
$\nu_{4})$ are located.

##  II Experiment

###  II.1 Experimental setup

Figure 1: Schematic drawing of the experimental apparatus. ADC: analog to
digital converter; BS: beam splitter; F.-P.: Fabry-Perot interferometer; M:
monochromator; P: photodiode; PC: desk-top computer.

Fig. 1 shows the experimental setup adopted for the WMS in this work. The
arrangement has been described in our previous work [9]. The source is a
Fabry–Perot type semiconductor heterostructure AlGaAs laser SHARP LT025MD that
emits a 40 mW continuous wave (cw) at 780 nm without any external optical
feedback. Thanks to its V-shaped junction cladding layers based on SHARP’s
original technology (VSIS chip structure) this type of cw diode laser has a
single transverse and single longitudinal mode. In this “free-running”
configuration the full width at half the maximum (FWHM) of its frequency
emission mode is around 20 MHz. Its operational current amplitude ranges
between 70 and 100 mA. The measurement cell containing the sample gas is a
multipass custom made Herriott type, 30 m path length. Another Herriott cell
contains water vapor for checking whether the obtained absorption features
come from H2O that could contaminate the cell. An iodine reference glass cell
is used for the precise wavenumber measurements. For the harmonic detection, a
sinusoidal current is mixed to the diode laser injection current and the
signals collected by pre-amplified silicon photodiodes are sent to the lock-in
amplifiers in order to extract the desired harmonic components. The methane
gas was supplied by Praxair Inc. with the following nominal characteristics:
grade 5.5 (purity $\geqslant$ 99.9995%), H2O $\leqslant 2$ ppmv,
O${}_{2}\leqslant 0.5$ ppmv, N${}_{2}\leqslant 10$ ppmv, H${}_{2}\leqslant
0.1$ ppmv.

###  II.2 Frequency modulation

The measured transmittance of the gas samples $\tau(\nu)$ is well described by
the Lambert-Beer equation:

$\tau(\nu)=e^{-\sigma(\nu)\,z}$ (1)

where $z=\rho\,l$ is the column amount (in molecule/cm2), i.e. the product of
the absorbing species number density $\rho$ (in molecule/cm3) and the optical
path $l$ (in cm) of the radiation through the sample; the absorption cross
section $\sigma(\nu)$ is expressed in cm2/molecule. In case of small optical
depths [$\sigma(\nu)z\ll 1$], always verified in our case, Eq. (1) can be
approximated by

$\tau(\nu)\simeq 1-\sigma(\nu)\,z\,.$ (2)

$\sigma(\nu)$ must take into account the shape of the absorption line:
Gaussian-like for Doppler, inhomogeneous broadening and Lorentzian-like for
collisional, homogeneous broadening. Other effects, like the Dicke narrowing
that occurs when the molecular mean free path is comparable to the wavelength
of the radiation [10], are not observed in our measurement conditions and are
not taken into account. For the purpose, the Voigt function well describes the
behavior of the absorption as a function of the radiation frequency:

$f(\nu)=\int_{-\infty}^{+\infty}{\frac{\exp{[-(t-\nu_{\circ})^{2}/{\it\Gamma_{\mathrm{G}}^{\mathrm{2}}}\ln
2]}}{(t-\nu)^{2}+{\it\Gamma_{\mathrm{L}}^{\mathrm{2}}}}}dt$ (3)

where $\nu_{\circ}$ is the gas resonance frequency, ${\it\Gamma_{\mathrm{G}}}$
and ${\it\Gamma_{\mathrm{L}}}$ are the Gaussian and the Lorentzian half-widths
at half the maximum (HWHM) respectively. We used the FM technique and
therefore the emission frequency $\bar{\nu}$ of the DL was sinusoidally
modulated at the frequency $\nu_{\mathrm{m}}=\omega_{\mathrm{m}}/2\pi$,
resulting in

$\nu=\bar{\nu}+a\cos\omega_{\mathrm{m}}t\,.$ (4)

In this case the transmitted intensity depends on both the line shape and the
modulation parameters and can be written as a cosine Fourier series:

$\tau(\bar{\nu}+a\,\cos\omega_{\mathrm{m}}t)=\sum_{n=0}^{\infty}H_{n}(\bar{\nu},a)\,\cos
n\omega_{\mathrm{m}}t$ (5)

where $H_{n}(\bar{\nu})$ is the $n$-th harmonic component of the modulated
signal. By using a lock-in amplifier tuned to a multiple $n\nu_{\mathrm{m}}$
($n=1,2,...$) of the modulation frequency, the output signal is proportional
to the $n$-th component $H_{n}(\bar{\nu})$ and when the amplitude $a$ is
chosen smaller than the width of the transition line, the $n$-th Fourier
component is proportional to the $n$-order derivative of the original signal:

$H_{n}(\bar{\nu},a)=\frac{2^{1-n}}{n!}\,a^{n}\,\left.\frac{d^{n}\tau(\nu)}{d\nu^{n}}\right|_{\nu=\bar{\nu}},\quad\quad
n\geqslant 1\,.$ (6)

For pressure broadening measurements we used a low modulation amplitude and we
detected the second harmonic component (2$f$ detection), therefore the output
signal was proportional to the second order derivative of the real absorption
line. This enhanced the signal-to-noise (S/N) ratio and reduced to zero the
unwanted background. In order to extract the line parameters, we used a
nonlinear least-squares fit procedure explained elsewhere [11]. In particular
we fitted the full-width at half the maximum (FWHM) $\gamma_{\mathrm{L}}$ vs
pressure by the general expression:

$\gamma_{\mathrm{L}}(p)=2{\it\Gamma_{\mathrm{L}}}(p)=\gamma_{i}p_{i}+\gamma_{\mathrm{self}}p_{\circ}$
(7)

where $\gamma_{i}$ is the FWHM broadening coefficient related to the $i$
buffer gas, $p_{i}$ is the buffer gas partial pressure,
$\gamma_{\mathrm{self}}$ is the sample gas FWHM self-broadening coefficient
and $p_{\circ}$ is the sample gas partial pressure.

###  II.3 High modulation regime

To obtain the line positions even for the weakest lines it was necessary to
use large values of the modulation amplitude. This substantially improves the
S/N ratio, but when $a$ is increased, the derivative approximation of Eq. (6)
fails and the $n-$th harmonic component $H_{n}(\nu,a)$ becomes [12]

$H_{n}(\nu,a)=\frac{2}{\pi}\int_{0}^{\pi}\tau(\nu+a\cos\theta)\cos
n\theta\;d\theta\,.$ (8)

The analytical evaluation of this integral is not always possible. Arndt [13]
and Wahlquist [14] derived the analytical form of the harmonic components for
a Lorentzian absorption function. The expression for the $n-$th harmonic
component can be obtained by inverting Eq. (5):

$H_{n}(x,m)=\varepsilon_{n}\,\mathrm{i}^{n}\,\int_{-\infty}^{+\infty}\hat{\tau}(\omega)\,J_{n}(m\omega)\,e^{\mathrm{i}\omega
x}\,d\omega$ (9)

where

$\hat{\tau}(\omega)=\frac{1}{2\pi}\int\tau(x)\,e^{-\mathrm{i}\omega x}\,dx$
(10)

is the Fourier transform of the transmittance profile; $x=\nu/{\it\Gamma}$ and
$m=a/{\it\Gamma}$ are respectively the frequency and the amplitude of the
modulation, normalized to the line-width ${\it\Gamma}$; $J_{n}$ is the $n-$th
order Bessel function; $\varepsilon_{0}=1$, $\varepsilon_{n}=2$
($n=1,2,\cdots$) and i is the imaginary unit. Assuming a Lorentzian absorption
line-shape centered at $\nu=0$ (this is acceptable when, as in our case,
collisional broadening dominates) the cross-section coefficient will be:

$\sigma_{\mathrm{L}}(x,m)\propto\frac{1}{1+(x+m\ cos\omega t)^{2}}$ (11)

Referring to the work of Arndt we recalculated the second Fourier component of
the cross-section coefficient by putting $n=2$:

$H_{2}(x,m)=-\frac{1}{m^{2}}\biggl{[}\frac{\\{[(1-\mathrm{i}x)^{2}+m^{2}]^{1/2}-(1-\mathrm{i}x)\\}^{2}}{[(1-\mathrm{i}x)^{2}+m^{2}]^{1/2}}+\mathrm{c.c.}\biggr{]}$
(12)

and by eliminating the imaginary part:

$H_{2}(x,m)=\frac{2}{m^{2}}-\frac{2^{1/2}}{m^{2}}\times\frac{1/2[(M^{2}+4x^{2})^{1/2}+1-x^{2}][(M^{2}+4x^{2})^{1/2}+M]^{1/2}+|x|\,[(M^{2}+4x^{2})^{1/2}-M]^{1/2}}{(M^{2}+4x^{2})^{1/2}}$
(13)

where

$M=1-x^{2}+m^{2}\,.$

Fig. 2 shows the behavior of Eq. (13) as a function of the modulation
parameter $m$. It can be seen that it is proportional to the 2nd derivative of
the absorption feature only for low modulation; for $m=3$ it is completely
deformed by broadening, as it happens in the reality.

Figure 2: Behavior of Eq. (13) as a function of the modulation parameter $m$.

This function, properly normalized, has been used to fit all the features
obtained in this work by the 2$f$ detection spectroscopy with high modulation
($m=2.2-2.3$).

##  III Experimental results

112 absorption lines have been observed in this work and their position
measured within 0.01 cm-1 by the comparison with a very precisely known I2
absorption spectrum [15] coming from the I2 reference cell. Table 1 shows the
results with the wavenumbers $\nu^{\prime}$ in vacuum. They presumably belong
to the combination overtone band $3\nu_{1}+\nu_{3}+(\nu_{2}$ or $\nu_{4})$ [5,
16] or perhaps $2\nu_{1}+\nu_{2}+2\nu_{3}$ [16], where $\nu_{1}$ is the
symmetric, $\nu_{3}$ is the asymmetric stretch and $\nu_{2}$ is the bending
fundamental vibrations. There are a few gaps in the Table ought to the lack of
emission wavelengths of the available laser diodes, therefore this Table does
not cover all the lines of the band but most of them. With the derivative
spectroscopy method adopted here it is difficult to obtain absolute intensity
measurements directly from the 2$f$ signals. We tried to measure the absolute
transmission values using the direct absorption (DA) technique by subtracting
the background, at room temperature (RT) on the same path-length as the one
adopted with WMS and with the CH4 pressure around 90 Torr that gives the best
S/N. The maximum absorption cross sections shown in the Tables for most of the
lines are obtained in this way.

Table 1: List of the observed CH4 lines along with the maximum absorption cross sections. $\nu^{\prime}$ (cm-1) | $\sigma_{\mathrm{max}}$ |  $\nu^{\prime}$ (cm-1) | $\sigma_{\mathrm{max}}$
---|---|---|---
| ($10^{-24}\frac{\mathrm{cm}^{2}}{\mathrm{molecule}}$) |  | ($10^{-24}\frac{\mathrm{cm}^{2}}{\mathrm{molecule}}$)
12836.84 | $0.14\pm 0.05$ | 12797.29 | $0.28\pm 0.05$
12812.43 |  | 12794.27 | $0.18\pm 0.05$
12812.34 | $0.10\pm 0.05$ | 12794.13 | $0.15\pm 0.05$
12812.14 |  | 12793.62 |
12811.78 | $0.31\pm 0.05$ | 12793.54 |
12809.62 |  | 12793.37 | $0.49\pm 0.05$
12809.53 |  | 12793.24 |
12809.43 | $0.33\pm 0.05$ | 12793.13 |
12807.80 | $0.67\pm 0.05$ | 12792.43 | $0.30\pm 0.05$
12807.71 |  | 12792.02 |
12806.52 | $1.06\pm 0.05$ | 12791.87 | $0.31\pm 0.05$
12804.95 | $0.68\pm 0.05$ | 12791.19 | $0.19\pm 0.05$
12804.73 |  | 12790.98 |
12804.60 | $0.17\pm 0.05$ | 12790.55 |
12803.50 | $0.26\pm 0.05$ | 12790.47 | $0.30\pm 0.05$
12803.08 | $0.29\pm 0.05$ | 12781.82 | $0.38\pm 0.05$
12801.96 | $0.08\pm 0.03$ | 12781.41 | $0.56\pm 0.05$
12801.84 | $0.48\pm 0.03$ | 12781.11 | $0.31\pm 0.05$
12800.69 | $0.33\pm 0.03$ | 12779.52 | $0.49\pm 0.05$
12799.09 | $0.42\pm 0.05$ | 12779.13 | $0.32\pm 0.05$
12798.98 |  | 12778.58 | $0.34\pm 0.05$
12798.94 |  | 12778.27 | $0.24\pm 0.05$
12798.63 |  | 12777.81 |
12798.56 | $0.21\pm 0.05$ | 12777.72 |
12798.39 | $0.18\pm 0.05$ | 12777.65 |
12798.22 |  | 12777.41 | $0.19\pm 0.05$
12798.10 |  | 12777.34 |
12797.98 |  | 12775.61 | $0.15\pm 0.05$
Table 1: List of the observed CH4 lines (continued). $\nu^{\prime}$ (cm-1) | $\sigma_{\mathrm{max}}$ |  $\nu^{\prime}$ (cm-1) | $\sigma_{\mathrm{max}}$
---|---|---|---
| ($10^{-24}\frac{\mathrm{cm}^{2}}{\mathrm{molecule}}$) |  | ($10^{-24}\frac{\mathrm{cm}^{2}}{\mathrm{molecule}}$)
12775.42 | $0.16\pm 0.05$ | 12756.65 |
12775.28 | $0.24\pm 0.05$ | 12756.36 | $1.0\pm 0.1$
12774.93 | $0.14\pm 0.05$ | 12755.48 |
12774.72 |  | 12755.36 |
12774.27 | $0.45\pm 0.05$ | 12755.13 | $0.41\pm 0.05$
12773.83 | $0.77\pm 0.05$ | 12755.00 |
12767.30 |  | 12753.52 | $0.79\pm 0.08$
12767.14 | $0.90\pm 0.06$ | 12753.44 |
12767.00 |  | 12752.18 |
12762.97 | $0.26\pm 0.05$ | 12752.08 |
12762.29 |  | 12751.08 | $1.0\pm 0.1$
12762.20 |  | 12749.46 | $0.25\pm 0.05$
12762.00 | $0.11\pm 0.05$ | 12749.12 |
12761.92 |  | 12746.48 | $0.50\pm 0.05$
12761.83 |  | 12746.05 | $0.31\pm 0.05$
12761.33 | $0.14\pm 0.05$ | 12742.99 | $0.33\pm 0.05$
12760.86 | $0.35\pm 0.05$ | 12742.21 | $0.30\pm 0.05$
12760.73 | $0.64\pm 0.05$ | 12741.22 | $0.22\pm 0.05$
12760.57 | $0.22\pm 0.05$ | 12740.15 | $0.39\pm 0.05$
12759.52 |  | 12737.32 | $0.16\pm 0.05$
12759.81 | $0.22\pm 0.05$ | 12729.71 | $0.4\pm 0.1$
12759.42 | $0.4\pm 0.1$ | 12729.43 | $0.17\pm 0.05$
12759.23 |  | 12721.87 | $0.20\pm 0.05$
12757.74 |  | 12714.64 |
12757.48 |  | 12714.48 | $0.55\pm 0.06$
12757.19 | $0.94\pm 0.05$ | 12713.30 | $0.92\pm 0.06$
12757.11 |  | 12712.19 | $0.27\pm 0.05$
12756.90 | $0.38\pm 0.05$ | 12695.17 | $0.22\pm 0.06$

An example of a crowded methane spectrum collected by using the WMS and the
2nd derivative technique is shown in Fig. 3, where the methane lines are
extracted by the fit procedure. Their positions are: 12793.13 cm-1, 12793.24
cm-1, 12793.37 cm-1, 12793.54 cm-1 and 12793.62 cm-1. The I2 absorption line
registered contemporary with methane is shown in dotted line. Its position is
known to be $(12793.5780\pm 0.0017)$ cm-1 from the I2 absorption lines
database [15]. When dealing with free-running diode lasers, a frequency sweep
of their emission is always associated with a variation of the emission
intensity, therefore in Fig. 3 the relative intensity ratios does not give the
real ratios as they suffer this intensity change; in particular as the
wavenumber increases the intensity of the DL mode decreases. In the range of
values of the Figure the DL mode intensity changes by 35%. In Table 1 the
cross-section is given only for the more intense of them.

Figure 3: 2nd derivative signal of the methane transmission spectrum around
781.4 nm (a) obtained by WMS with 10 Hz bandwidth at
p${}_{\mathrm{CH_{4}}}=84$ Torr and T = 296 K. The peaks extracted by the fit
procedure (b) are shifted down for clearness. The I2 reference signal (c) is
shown dotted.

The complexity of the structure of the overtone band does not permit an
immediate and correct quantum classification of the ro-vibrational resonances.
For these highly excited levels the numerous possible resonances between ro-
vibrational levels can modify significantly the intensity as well as the
position of the expected lines [17]. An aid to the classification job can come
from working at very low temperature ($\leqslant 20$ K) using the supersonic
methane jet expansion [18]. In this case only a few first rotational levels
will be populated and the level superpositions will be smaller or absent.

##  IV Line broadening measurements

Table 2 shows the measured CH4 pressure broadening coefficients by it-self,
air, H2 and He gases, at RT and only for a few more intense and well isolated
lines, where the modulation amplitude could be kept low to continue using the
second derivative approximation explained in Section II.2. The shown errors
are the maximum errors (3$\sigma$). The pressure line-shifting effect is
usually about an order of magnitude lower then the broadening, but the
sensitivity of this spectrometer does not permit to obtain reliable
measurements, therefore they are not reported here.

Table 2: Methane pressure broadening FWHM coefficients ($\gamma$) at room temperature. $\nu^{\prime}$ | $\gamma_{\mathrm{self}}$ | $\gamma_{\mathrm{air}}$ | $\gamma_{\mathrm{H_{2}}}$ | $\gamma_{\mathrm{He}}$
---|---|---|---|---
(cm-1) | (MHz/Torr) | (MHz/Torr) | (MHz/Torr) | (MHz/Torr)
12806.52 | $6.2\pm 0.1$ | $5.0\pm 0.1$ | $4.0\pm 0.3$ | $4.5\pm 0.4$
12804.95 | $7.1\pm 0.3$ |  |  |
12773.83 | $4.4\pm 0.1$ |  |  |
12767.14 | $3.7\pm 0.2$ |  |  |
12713.30 | $5.3\pm 0.1$ | $4.8\pm 0.4$ | $2.1\pm 0.2$ | $2.0\pm 0.2$

Examples of pressure broadening measurements are shown in Fig. 4 and Fig. 5,
where CH4 broadening behaviors by collision with it-self and helium buffer gas
respectively are displayed with their best linear fit for the 12713.30 cm-1
line.

Figure 4: FWHM self-broadening measurements for the 12713.30 cm-1 methane
absorption line as a function of pressure at room temperature.

Figure 5: FWHM He-broadening measurements for the 12713.30 cm-1 methane
absorption line as a function of pressure at room temperature.
P${}_{\mathrm{CH_{4}}}$ = 28 Torr.

For the 780 nm band there are no known previous pressure broadening
measurements, therefore a comparison can be done only with the results on
different wavelengths. Moreover, due to the lack of quantum assignment of the
lines in this band, only an average comparison can be done. Our broadening
coefficients at RT in average are similar or perhaps a little lower than ours
previous ones obtained at 840 nm [9] or to the ones coming from previous works
on fundamentals, overtones or combination of them [19, 20, 21, 22, 23, 24, 25,
26, 27, 28, 29, 30, 31, 32, 33] at RT. In the literature the self-broadening
coefficients got from the ro-vibrational bands in the NIR and MID IR span from
6 to 7 MHz/Torr [19, 30, 29, 31], while air-broadenings go from 4 to 6
MHz/Torr [20, 23, 32, 25, 33]. The H2-broadening coefficients are between 5
MHz/Torr [24] and 6 MHz/Torr [19]. The He-broadening coefficients got from the
literature [24, 25] for the $\nu_{4}$ and the $2\nu_{2}$ bands are in average
$\simeq 3.1-3.3$ MHz/Torr. From the HITRAN molecular spectroscopic database
[34] in the range $11200-11300$ cm-1 we found (4
$\leqslant\gamma_{\mathrm{air}}\leqslant 5.5$) MHz/Torr and (5.5
$\leqslant\gamma_{\mathrm{self}}\leqslant 6$) MHz/Torr, The self-broadening,
H2-broadening and He-broadening coefficients in the visible [35], at
combination overtones higher than ours, give in average 7, 6 and 4 MHz/Torr
respectively. In any case as the broadening coefficients $\gamma$ are very
dependent on the rotational quantum number $J$ ($\gamma$ values decrease as
$J$ increases), these results are purely indicative until their quantum
attribution will be clear.

##  V Conclusion

By using a tunable diode laser spectrometer with high resolving power
($\lambda/\Delta\lambda\approx 10^{7}$), the aid of the wavelength modulation
spectroscopy technique with the second harmonic detection and a 30 m total
path-length multipass measurement cell, 112 12CH4 absorption lines have been
detected around 12780 cm-1 and their positions measured within 0.01 cm-1. They
presumably belong to the combination overtone $3\nu_{1}+\nu_{3}+(\nu_{2}$ or
$\nu_{4})$ or $2\nu_{1}+\nu_{2}+2\nu_{3}$ ro-vibrational bands. The line
positions have been obtained by the comparison with reference I2 absorptions
and the utilization of a very precise atlas. The maximum absorption cross
section of the observed lines ranges between $1\times 10^{-25}$ and $1\times
10^{-24}$ cm2/molecule at room temperature. The collisional broadening
coefficients for different perturbing gases have been measured at RT for some
of the more intense lines and a comparison with similar data from the
literature has been tried.

##  VI Acknowledgments

The author is indebted to Davide Bertolini for the calculations in the high
modulation approximation. Thanks are due to Alessandro Barbini for the
electronic advising, Mauro Voliani for the mechanical assistance and Mauro
Tagliaferri for the technical support.

## References

  * Kuiper [1944] C. P. Kuiper, Titan: A satellite with an atmosphere, Astrophys. J. 100, 378 (1944).
  * Brown  _et al._ [2007] M. E. Brown, K. M. Barkume, G. A. Blake, E. L. Schaller, D. L. Rabinowitz, H. G. Roe, and C. A. Trujillo, Methane and ethane on the bright kuiper belt object, Astronom. J. 133, 284 (2007).
  * Karkoschka [1994] E. Karkoschka, Spectrophotometry of the jovian planets and titan at 300- to 1000-nm wavelength: The methane spectrum, Icarus 111, 174 (1994).
  * Vedder and Mecke [1933] H. Vedder and A. Mecke, Das rotationsschwingungsspectrum des methans, Z. Phys. 86, 137 (1933).
  * Giver [1978] L. P. Giver, Intensity measurements of the CH4 bands in the region 4350 Å to 10,600 Å, Can. J. Phys. 19, 311 (1978).
  * Dick and Fink [1977] K. A. Dick and U. Fink, Photoelectric absorption spectra of methane CH4, methane and hydrogen H2 mixtures, and ethane $\mathrm{C_{2}H_{6}}$, J. Quant. Spectrosc. Radiat. Transfer. 18, 433 (1977).
  * Singh and O’Brien [1994] K. Singh and J. J. O’Brien, Laboratory measurements of absorption coefficients for the 727 nm band of methane at 77 K and comparison with results derived from spectra of the giant planets, J. Mol. Spectrosc. 54, 607 (1994).
  * O’Brien and Cao [2002] J. J. O’Brien and H. Cao, Absorption spectra and absorption coefficients for methane in the 750–940 nm region obtained by intracavity laser spectroscopy, J. Quant. Spectrosc. Radiat. Transfer. 75, 323 (2002).
  * Lucchesini and Gozzini [2007] A. Lucchesini and S. Gozzini, Methane diode laser overtone spectroscopy at 840 nm, J. Quant. Spectrosc. Radiat. Transfer. 103, 209 (2007).
  * Dicke [1953] R. H. Dicke, The effect of collision upon the doppler width of the spectral lines, Phys. Rev. 899, 472 (1953).
  * De Rosa  _et al._ [1998] M. De Rosa, A. Ciucci, D. Pelliccia, C. Gabbanini, S. Gozzini, and A. Lucchesini, On the measurement of pressure induced shift by diode lasers and harmonic detection, Opt. Comm. 147, 55 (1998).
  * Webster  _et al._ [1988] C. R. Webster, R. T. Menzies, and E. D. Hinkley,  _Infrared laser absorption: Theory and applications. In: Measures RM, editor. Laser remote chemical analysis_ (John Wiley & Sons, Ltd, 1988).
  * Arndt [1965] R. Arndt, Analytical line shapes for lorentzian signals broadened by modulation, J. Appl. Phys. 36, 2522 (1965).
  * Wahlquist [1961] H. Wahlquist, Modulation broadening of unsaturated lorentzian lines, J. Chem. Phys. B 35, 1708 (1961).
  * Gerstenkorn  _et al._ [1982] S. Gerstenkorn, J. Verges, and J. Chevillard,  _Atlas du spectre d’absorption de la molécule d’iode: 11.000 cm -1 – 14.000 cm-1_ (Edition du CNRS, Laboratoire Aimé Cotton, Orsay, France, 1982).
  * Herzberg [1945] G. Herzberg,  _Molecular spectra and molecular structure_ (D. Van Nostrand Company, Inc., New York, NY, USA, 1945) Chap. II: Infrared and Raman spectra of polyatomic molecules, p. 308.
  * Halonen [1997] L. Halonen, Internal coordinate hamiltonian model for fermi resonances and local modes in methane, J. Chem. Phys. 106, 831 (1997).
  * Campargue  _et al._ [1995] A. Campargue, D. Permogorov, and R. Jost, Intracavity absorptions spectroscopy of the third stretching overtone transition of jet cooled methane, J. Chem. Phys. 102, 5910 (1995).
  * Varanasi [1971] P. Varanasi, Collision-broadened half-widths and shapes of methane lines, J. Quant. Spectrosc. Radiat. Transfer. 11, 1711 (1971).
  * Varanasi [1974] P. Varanasi, Air-broadened line widths of methane at atmospheric temperature, J. Quant. Spectrosc. Radiat. Transf. 15, 281 (1974).
  * Fink  _et al._ [1977] U. Fink, D. C. Benner, and K. A. Dick, Band model analysis of laboratory methane absorption spectra from 4500 to 10500 Å, J. Quant. Spectrosc. Radiat. Transfer. 18, 447 (1977).
  * Fox  _et al._ [1988] K. Fox, D. E. Jennings, E. A. Stern, and R. Hubbard, On the $4^{1}\Sigma^{+}$ state of the KCs molecule, J. Quant. Spectrosc. Radiat. Transfer. 39, 473 (1988).
  * Rinsland  _et al._ [1988] C. P. Rinsland, V. Malathy Devi, M. A. H. Smith, and D. C. Benner, Measurements of air-broadened and nitrogen-broadened lorentz width coefficients and pressure shift coefficients in the $\nu_{4}$ and $\nu_{2}$ bands of $\mathrm{{}^{12}CH_{4}}$, Appl. Opt. 27, 631 (1988).
  * Varanasi [1989] P. Varanasi, Measurements of collision-broadened line widths in the $\nu_{4}$-fundamental band of 12CH4 at low temperatures, J. Quant. Spectrosc. Radiat. Transf. 72, 335 (1989).
  * Zéninari  _et al._ [2001] V. Zéninari, B. Parvitte, D. Courtois, V. A. Kapitanov, and Y. N. Ponomarev, Measurements of air and noble-gas broadening and shift coefficients of the methane $\mathrm{R3}$ triplet of the $2\nu_{3}$ band, Appl. Phys. B 72, 953 (2001).
  * Gharavi and Buckley [2005] M. Gharavi and S. G. Buckley, Diode laser absorption spectroscopy measurement of linestrenghts and pressure broadening coefficients of the methane $2\nu_{3}$ band at elevated temperatures, J. Mol. Spectrosc. 229, 78 (2005).
  * Predoi-Cross  _et al._ [2005] A. Predoi-Cross, L. R. Brown, V. Malathy Devi, B.-T. M., and D. C. Benner, Multispectrum analysis of $\mathrm{{}^{12}CH_{4}}$ from 4100 to 4635 cm-1: 1. self-broadening coefficients (widths and shifts), J. Mol. Spectrosc. 232, 231 (2005).
  * Mondelain  _et al._ [2005] D. Mondelain, P. Chelin, A. Valentin, D. Hurtmans, and C. Camy-Peyret, High-resolution laser spectroscopy of the X${}^{1}\Sigma^{+}$ and (1)${}^{3}\Sigma^{+}$ states of 23Na85Rb molecule., J. Mol. Spectrosc. 233, 23 (2005).
  * Menard-Bourcin  _et al._ [2007] F. Menard-Bourcin, J. Menard, and C. Boursier, Temperature dependence of rotational relaxation of methane in the $2\nu_{3}$ vibrational state by self- and nitrogen-collisions and comparison with line broadening measurements, J. Mol. Spectrosc. 242, 55 (2007).
  * Boussin  _et al._ [1998] C. Boussin, L. Regalia, J. J. Plateaux, and A. Barbe, Line intensities and self-broadening coefficients for methane lines between 5500 and 6180 cm-1 retrieved with a multispectrum fitting technique, J. Mol. Spectrosc. 191, 381 (1998).
  * Lyulin  _et al._ [2011] O. M. Lyulin, V. I. Perevalov, I. Morino, T. Yokota, R. Kumazawa, and T. Watanabe, Measurements of self-broadening and self-pressure-induced shift parameters of the methane spectral lines in the 5556-6166 cm-1 range, J. Quant. Spectrosc. Radiat. Transfer. 112, 531 (2011).
  * Benner  _et al._ [2002] D. C. Benner, V. M. Devi, M. A. H. Smith, and C. P. Rinsland, Air-, N2-, and O2-broadening and shift coefficients in the $\nu_{3}$ spectral region of 12CH4, J. Quant. Spectrosc. Radiat. Transfer. 50, 65 (2002).
  * Hongliang  _et al._ [2016] M. Hongliang, C. Qiang, L. Zhensong, C. Weidong, V. Aurore, H. Yinbo, Z. Wenyue, G. Xiaoming, and R. Ruizhong, N2\- and air-broadened lorentz half-width coefficients of methane transitions around 3.38 $\mu$m using tdls, J. Quant. Spectrosc. Radiat. Transfer. 171, 50 (2016).
  * I. Gordon, L. Rothman, R. Hargreaves, R. Hashemi, E. Karlovets, F. Skinner, E. Conway, et al. [2022] I. Gordon, L. Rothman, R. Hargreaves, R. Hashemi, E. Karlovets, F. Skinner, E. Conway, et al., The HITRAN2020 molecular spectroscopic database, J. Quant. Spectrosc. Radiat. Transfer. 277, 107949 (2022).
  * Keffer  _et al._ [1986] E. C. Keffer, C. P. Conner, and W. H. Smith, Pressure broadening of methane lines in the 6190 Å and 6825 Å bands at room and low temperatures, J. Quant. Spectrosc. Radiat. Transfer. 35, 495 (1986).