# Absorption line broadening in atomic beams produced in a molecular beam epitaxy environment

Roman Rousseau, Jérôme Morville, Claude Botella, Guillaume Saint-Girons

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Roman Rousseau, Jérôme Morville, Claude Botella, Guillaume Saint-Girons. Absorption line broadening in atomic beams produced in a molecular beam epitaxy environment. Optics Express, 2024, 32 (23), pp.40202. 10.1364/oe.536538. hal-04776857

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# Absorption line broadening in atomic beams produced in a molecular beam epitaxy environment

ROMAN ROUSSEAU,<fnref n="1" /> JÉRÔME MORVILLE,<fnref n="2" /> CLAUDE BOTELLA,<fnref n="1" /> AND GUILLAUME SAINT-GIRONS<fnref n="1" />,∗

**Abstract:** Tunable diode laser absorption spectroscopy (TDLAS) is used to measure the 6s<fnref n="2" /> ¹S-5d6p ³D⁰ absorption line profile of a Ba atomic beam produced in a molecular beam epitaxy (MBE) reactor. Despite the noisy MBE environment, a signal-to-noise ratio up to 100 is obtained thanks to a thorough optimization of the measurement setup. A model that realistically describes this absorption profile is presented, taking into account the angular distribution of atomic concentration in the atomic beam as well as the reactor and setup geometry. Our results highlight the influence of the latter on the Doppler line broadening, and allows to estimate the angular distribution of concentration in the beam emitted by the effusion cell. The model can be adapted to estimate the absorption profile for any element in any reactor geometry, making it a useful tool to design MBE dedicated atomic absorption sensors such as optical flux monitoring sensors, or to characterize the beam emitted by any effusion cell.

Published by Optica Publishing Group under the terms of the [Creative Commons Attribution 4.0 License](https://creativecommons.org/licenses/by/4.0/). Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

## 1. Introduction

Since the pioneering works published at the beginning of the 20th century [1,2], molecular beams have been produced to study the structure and properties of molecules or have enabled considerable advances in various domains such as quantum chemistry, condensed matter, few-body physics, atomic clocks and metrology, etc [3–11]. For most of these spectroscopy applications, and specifically for the flagship applications that are high resolution spectroscopy and study of reaction dynamics, the desired properties for molecular beams are a controlled (possibly low) molecular velocity, a narrow molecular velocity distribution along the optical axis (i.e a low beam divergence/ quasi-collimated beam) and a low beam temperature [3,11]. Indeed, in such beams, all the atoms/molecules are in their lowest energy states which reduces spectral congestion and gives access to longer light-matter interaction times, which increases measurement resolution [12–15]. Moreover, low beam temperatures lead to low vibrational-rotational disorder, considerably simplifying the absorption spectrum. Besides, studying molecular scattering requires narrow and tunable molecule velocity distribution [16–18]. In particular, quantum effects in molecular scattering are only accessible in molecular beams with low molecular velocity [19–22].

In the context described above, the techniques developed to generate molecular beams for spectroscopy have been optimized for more than a century to obtain low beam temperature (< 1 K), low controllable molecular velocity and a low dispersion of the molecule velocity distribution. Two main techniques exist to generate such beams, namely the supersonic expansion [23], and the use of Knudsen effusion sources [24]. In supersonic expansion sources, the internal energy of a relatively high-pressure vapor stored in a container and put into contact with vacuum via a small aperture is adiabatically converted into kinetic energy by the expansion of the vapor in the

<footnote n="1">¹INL-UMR5270/CNRS, Ecole Centrale de Lyon, 36 avenue Guy de Collongue 69134 Ecully cedex, France</footnote>

<footnote n="2">²ILM-UMR5306/CNRS, UCBL, 10 rue Ada Byron 69622 Villeurbanne cedex, France</footnote>

<footnote>∗guillaume.saint-girons@ec-lyon.fr</footnote>
vacuum. Supersonic beams are naturally very cold (typically <1 K) but present high molecular velocity, that can be reduced down to almost zero using various strategies [14,25,26]. The operating principle of effusion sources is comparable, except that the pressure in the container is low, so that the molecule mean free path is large (collision free regime). Filtering strategies and progress in effusion source design have allowed to reach beam temperatures as low as a few kelvin and sub 100 m/s molecule velocities [27–29].

Molecular beams are also used in a completely different context, namely that of the growth of thin layers under ultra-high vacuum, in molecular beam epitaxy reactors (MBE). They are produced using MBE effusion sources, whose design sensibly differs from that of Knudsen effusion sources [30]. While the optical properties (atomic/molecular absorption/emission) of the molecular beams for spectroscopy are well known and studied for a long time, those of MBE molecular beams are much less so. The few studies available in literature do neither extensively analyze the line Doppler broadening, nor the way in which the angular distribution of atomic concentration in the beam influences the absorbance [31–36]. Yet, studying the absorption profiles of MBE molecular beams is interesting in several respects. First, the angular distribution of atomic concentration in MBE molecular beams influences the growth process, and in particular the homogeneity of deposit thickness and composition. As we show in this work, absorption spectroscopy is a powerful tool to characterize this distribution, and could contribute to the optimization of MBE effusion source design. Second, so-called optical flux monitoring systems (OFM), used to monitor the growth rate in MBE reactors, are based on measuring the optical absorption of MBE molecular beams [37]. These systems use hollow cathode lamps (HCL) as light sources, as these atomic lamps emit very narrow discrete lines, the spectral width of which (in the GHz range) is expected to be in the same order of magnitude as that of the MBE molecular beam absorption lines. Although it is suspected that the spectral mismatch between HCL emission and MBE molecular beam absorption affect OFM sensors [31,38], this effect has never been studied, probably due to the lack of data concerning the absorption spectrum of MBE molecular beams.

Certainly, the vast literature concerning molecular beams for spectroscopy provides a rough idea of the absorption properties of MBE molecular beams. However, the latter are not optimized to minimize the atomic/molecular temperature, nor to control the atomic/molecular velocity or to minimize the velocity dispersion. On the contrary, the angular divergence of the beam produced using MBE effusion sources is deliberately large, of the order of a few tens of °, to ensure good homogeneity of the growth rate over the entire substrate. Moreover, MBE effusion sources are not equipped with systems to cool the molecular beam, the temperature of which is therefore equal to the temperature of the source, i.e. typically ranging from a few hundreds °C to 2000°C depending on the element to be evaporated. Finally, MBE reactor geometry imposes that the optical probe beam used in OFM systems is not orthogonal to the molecular beam to be probed, contrasting with the standard conditions used in experiments involving molecular beams for spectroscopy. These elements make it difficult to use the data available in the literature concerning the latter to reliably anticipate the absorption properties of MBE molecular beams.

In this context, the present work reports a study of the absorption profile of a Ba atomic beam produced in a MBE environment, probed using tunable diode laser absorption spectroscopy (TDLAS) [39]. On the one hand, Ba is well suited to study the line broadening mechanisms in a MBE reactor. In fact, the main Ba isotope does not possess a nuclear spin so that its optical absorption spectrum, composed of single isolated transitions, is not complexified by a hyperfine structure. On the other hand, the TDLAS technique with a spectral resolution ultimately limited by the laser linewidth (in the MHz range) enables to probe with high accuracy such narrow profiles, as expected for the Ba atomic beam (expected linewidth in the GHz range). The non-orthogonality of the atomic beam with the optical axis induces a global Doppler shift of the line, that we determine using a double optical pass configuration, which suppresses the need
for a sophisticated absolute frequency reference system. We present a model that realistically describes the experimental results. This model accounts for the reactor geometry, the velocity distribution in the beam, and the atomic concentration profile described using different types of functions. Fitting the model to the experimental data allows to determine which of the functions best describe the atomic concentration profile. These results provide quantitative information on the angular distribution of Ba atomic concentration in the beam. Our approach, adaptable to any reactor geometry, can be applied to optimize MBE effusion source design. Our work also highlights important elements for the design of OFM systems, showing in particular that the absorption line profile is very dependent on the reactor geometry, and possibly significantly different from HCL emission line profile.

## 2. Experimental

The heart of the experimental setup is a RIBER C21 MBE reactor, dedicated to the growth of oxide thin layers and in particular ferroelectric $BaTiO_3$. Two optical platforms are attached on both sides of the reactor, as sketched in Fig. 1. Elemental atomic effusion cells, directed toward the 2'' substrate holder, are mounted in the lower part of the reactor. The latter is equipped with a primary pump, a turbomolecular pump and a cryogenic panel filled with liquid nitrogen so that the background pressure is in the $10^{-10}$ Torr range. This corresponds to a typical mean free path of several hundred kms (for $O_2$). In such conditions, applying a controlled electrical heating to the cylindrical crucible containing the material in the cell generates a diverging effusive atomic beam, whose diameter is large enough to ensure a fairly homogenous deposition onto the rotating substrate. The distance between the cell crucible aperture and the substrate is around 19 cm, and the crucible depth in 9 cm, so that the distance between the Ba metallic load and the substrate approximately ranges from 28 to 19 cm. Two broadband AR coated optical ports allow the light to pass about 2 cm below the substrate.

<figure><img src="image_2.png" /><figcaption>**Fig. 1.** Schematic of the experimental setup showing the two optical systems mounted on both sides of the MBE reactor and the electronics and acquisition system.</figcaption></figure>

The optical setup was designed to measure the $6s^2 \ 1S-5d6p \ 3D^0$ Ba absorption line, centered at 413.25 nm. This measure was performed for Ba cell temperatures ranging from 530 to 600°C, which typically corresponds to $BaTiO_3$ growth rates of a few monolayers/min.

The optical system on the left in Fig. 1, onto which the laser source is mounted, is a 60 × 60 cm aluminum breadboard attached to the reactor chassis, just in front of one of the optical ports. On
the opposite port, the optical system consists in a cage system in which a beam splitter, a mirror and an optical detection system are mounted. All the optical parts thus share the vibrations of the reactor, mainly caused by the turbomolecular pump.

The laser source is an external cavity diode laser (ECDL) (Toptica DL100) emitting between 411 nm and 415 nm. The ECDL temperature is stabilized at 20°C using a peltier cooler and the emission wavelength can be tuned by rotating the laser external grating. A fine-pitch screw is used for coarse tuning while fine tuning is realized with a piezo actuator. However, single mode emission requires a perfect frequency match between one of the external cavity modes and a mode of the laser diode chip. The latter can be tuned by adjusting the diode current. The laser driver (Toptica DC110) allows to apply synchronous temporal signals onto both the piezo driver and on the laser current, yielding a mode-hope free continuous scan over more than 30 GHz.

The elliptical output beam of the ECDL is firstly passed through a pair of anamorphic prisms to obtain a circular beam and then send through a thick glass beam splitter (BS1). One of the reflected beams (~4% of the incident power) is sent onto an adjustable gain Si-photodiode PD0 (Thorlabs PDA36A) to monitor the laser power during the frequency scan. The other reflected beam (also near 4% of the incident power) is sent onto a 10 cm long solid glass cavity producing a low-finesse etalon signal with a free spectral range (FSR) of 1 GHz. In order to optimize the fringe contrast, this etalon signal is measured in reflection on the other side of BS1 using the PD2 photodiode PD2 (Thorlabs PDA36A). This signal is used to determine the relative frequency scale of the recorded spectra. The beam transmitted through the etalon is sent to a wavelength meter (Coherent WaveMaster) with an absolute resolution of 1pm, sufficient to bring the laser wavelength near the Ba transition with the coarse adjustment. It can be noted that no optical isolator is used at the laser exit. Instead, all optical surfaces normal to the beam are slightly tilted to eliminate any optical feedback that would disturb the laser frequency and power. Especially, this tilt slightly modifies the etalon FSR. However, we estimate an FSR variation lower than 1kHz, using the distance from the laser to the etalon (~1 m) and the laser beam diameter (<1 mm). This variation is negligible compared to the spectral resolution of the measurement, ultimately fixed by the laser linewidth (around 1 MHz). With a set of three mirrors, the forward beam is then injected through the two MBE reactor optical ports to reach the 90/10 beam-splitter BS2 mounted on the output optical platform. The transmitted beam (10%) is collected using the PD3 photodiode (Thorlabs PDA36A), and the reflected beam (90%) is reflected by a high-reflection mirror close to normal incidence. This results in a backward beam, with roughly 80% of the forward beam power, travelling the MBE reactor in the opposite way. Its power is measured using the PD1 photodiode PD1 (Thorlabs PDA36A). Lastly, a variable optical density filter is inserted on the forward beam to reduce the laser power and to prevent absorption saturation.

The signals recorded by the four photodiodes are sent to a 16-bit analog-digital-converter (ADC) (National Instrument, USB-6251) with a sample rate of 1 MS/s. A function generator sends a sine signal with a frequency of 10 Hz to the laser driver and a synchronous TTL signal to trig the ADC. The amplitude of the sine signal is adjusted for a continuous mode-hop-free frequency scan of typically 20 GHz. Data are processed in a Labview software environment. The acquisition temporal window corresponds to a half-period of the sine signal, the other half being used for data-treatment. The data temporal scale is first converted into a relative frequency scale using a zero-crossing algorithm applied to the derivative of the etalon signal (Fig. 2(a) and 2(b)) and the etalon FSR (1 GHz). PD3 and PD1 signals are then normalized by the PD0 signal to suppress the noise resulting from laser power fluctuations. The resulting normalized signal from PD3 can be expressed as $S_3(\nu) = G_3 \cdot (T_{BS2}T_{et}/R_{et}) \cdot e^{-\alpha_F(\nu)L}$, and that from PD1 as $S_1(\nu) = G_1 \cdot (R_{BS2}^2 T_{et}/R_{et}) \cdot e^{-\alpha_F(\nu)L} \cdot e^{-\alpha_B(\nu)L}$, where $G_i$ is the gain of PDi, $R_{et}$ and $T_{et}$ are the effective reflection and transmission coefficients of BS1, $T_{BS2}$ and $R_{BS2}$ are respectively the transmission and reflection coefficients of BS2, $\alpha_F(\nu)$ is the absorption coefficient per unit length of the forward beam interacting over the length $L$ with the atomic beam, and $\alpha_B(\nu)$ is
the absorption coefficient per unit length of the backward beam. The interaction length *L* is considered the same for the two directions as the angle between the two beams is less than 5 mrad. These two signals are averaged over time. Taking the natural logarithm, the absorbance spectrum is obtained after offset subtraction to take into account the optical power and the photodiode gain difference. However, typical baselines obtained this way show a marked polynomial dependency (Fig. 2(c)) that we attribute to the different optical paths (and coatings) encountered by each of the beams. We correct this effect by fitting the data outside of the absorbance profile using a 5th-order polynomial, and subtracting this background to obtain the absorbance spectra of the two beams with a flat zero baseline (Fig. 2(d)). Since the signal S1 measures the contribution of the two passes into the atomic beam, the absorbance spectra corresponding to the backward direction solely (brown line) is then simply obtained by subtraction of the forward pass (blue line) from the double pass (grey line).

<figure><img src="image_3.png" /><figcaption>**Fig. 2.** (a) Temporal traces of PD1 and PD3 photodiodes, and (b) derivative of the etalon signal from PD2 photodiode. The peak positions of the etalon signal are determined with a zero-crossing algorithm applied to its derivative signal, and are used to determine a relative frequency scale. (c) Spectra of the natural logarithm of the normalized signals corresponding to the forward pass (blue) and the double pass (grey). The baselines (black dotted lines) are fitted outside of the absorbance profile using a 5th-order polynomial function. (d) Spectra obtained after baseline subtraction for the forward (blue) and double (grey) pass. The effusion cell temperature is 570 °C.</figcaption></figure>

This subtraction is applied in real time during the averaging process. We determine the optimal number of averaged spectra by computing the Allan-Werle deviation [40] which determines the time window over which white noises dominate. Optimizing the signal to noise ratio (SNR) of an absorbance profile requires maximizing the accuracy on both the intensity and the frequency scales of the spectra. The absorbance scale noise is governed by the photodiodes noise, whereas the frequency scale noise is mainly induced by the noise of the frequency tuning and ultimately by the laser linewidth on the measurement time. We thus evaluate the Allan-Werle deviation on three points of the spectrum, namely the baseline, the top and the side of the absorbance profile (Fig. 3). This last point is indeed a combination of both types of noise and will fix the optimal number of averaging. It indicates that white noise dominates over tens of seconds enabling to increase the absorbance spectrum SNR by more than a factor ten by averaging 100 samples.

Relevant assessment on the influence of the atomic beam geometry on the absorption line shape requires the laser emission intensity to be small enough to prevent any saturation effect. The laser beam cross section is constrained to be lower than 1 mm² by the imposed small angle of the double pass geometry and the laser output power is close to 10 mW. This is sufficient to saturate the transition as visible on Fig. 4, where the measured absorbance decreases with
<figure><img src="image_4.png" /><figcaption>**Fig. 3.** Allan-Werle deviation plot of the backward absorbance recorded at 3 different locations (Baseline, Top, Side) of the absorption, as shown in the inset. The effusion cell temperature was of 570 °C. The white noise dominates over roughly ten seconds enabling to average 100 spectra. The red dashed line indicates the $1/\sqrt{N_s}$ law with $N_s$ the number of samples.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>**Fig. 4.** Forward (blue shades) and backward (red shades) absorbance for different laser power obtained by inserting optical density filters (OD) at the entrance of the reactor (attenuation factor $10^{-OD}$). The effusion cell temperature is 550 °C.</figcaption></figure>

decreasing laser beam attenuation. As the backward beam power is reduced by the two reflection on BS2, the saturation is more pronounced on the forward absorbance. A laser power attenuation of roughly 30 is required to prevent any saturation but at this power level, the signals start to be significantly affected by photodiode noise. As this configuration is not optimal from the point of view of SNR, the forward and backward absorption profiles were also measured separately
<figure><img src="image_6.png" /><figcaption>**Fig. 5.** Forward absorbance (a) and backward absorbance (b) measured separately (by injecting the laser probe beam from the input for the forward absorption and from the output port for the backward absorption profile) for different effusion cell temperatures. This optimized configuration enables to reach an absorbance baseline noise with a standard deviation (SD) of $2.10^{-4}$. (c) Forward (blue line) and backward (red line) absorbance profiles measured separately superimposed to the profiles with lower SNR measured simultaneously (black lines). The Ba cell temperature was of 570°C. Pink line: rest line center position, deduced from the fit of the absorption profile (section 3). Dashed black lines: typical estimated HCL line profiles (Gaussian shaped, full width at half maximum 1.5 and 2.5 GHz).</figcaption></figure>

by injecting the laser probe beam from the input port for the forward absorption profile (BS2 is removed), and from the output port for the backward absorption profile. The results are shown on Fig. 5(a) and 5(b) where the two absorbance spectra were also measured as a function of the effusion temperature cell. The absorbance baseline noise indicates a standard deviation of $2.10^{-4}$ after averaging and a SNR level up to 100 is demonstrated, which is better than similar works reported in the literature, such as in [36]. The absorbance increases with the cell temperature as expected from the dependency of the effusive flow rate with temperature.
While a common frequency scale can be defined for the spectra measured simultaneously, this cannot be done for the spectra measured separately in the forward and backward directions. We thus plot these profiles with improved SNR in a common consistent frequency scale by superimposing them to those measured simultaneously, as shown in Fig. 5(c)).

The shape of these Doppler-broadened profiles is consistent with the geometry of the system. As the probe laser beam is not orthogonal to the atomic beam, the forward and backward profiles are not superimposed, and as the atomic beam is also divergent, the absorption profiles are asymmetric. The spectra consist of a main absorption feature corresponding to the most abundant $^{138}$Ba isotope, and a secondary peak, clearly visible on the backward pass and as a discrete shoulder on the forward pass (Fig. 5(c)). It is difficult to attribute this secondary peak to a specific isotope. Indeed, Barium possesses several stable isotopes with comparable natural abundance, some of them with a hyperfine structure. This complexity and the lack of literature data on Ba isotopic shift make it difficult to assign the secondary peak to specific isotopes. As a consequence, it is described in the model presented in the next section as a single isolated transition. The main peak has a full width at half maximum (FWHM) in the 500 MHz range, approximately ten times larger than the Barium natural linewidth $\Gamma_N = 58.75$ MHz [41] due to the Doppler broadening.

These measurements provide interesting information for the design of OFM sensors. HCL are used as light sources in these sensors, as mentioned above. The typical emission profiles of these sources (dashed lines in Fig. 5(c)) have spectral widths in the 1.5 to 2.5 GHz range, depending on the element and on the lamp current [42,43]. These typical widths are three to five times larger than the our measured FWHM, confirming the conclusions of some earlier works [31,42]. In HCL-based OFM sensors, a significant fraction of the incident light is not absorbed by the atomic beam, which reduces the absorption contrast thus affecting the sensor sensitivity, and causing non-linearity in their response. Our results also evidence the interest of two-passes OFM sensors to improve the HCL/absorption line spectral overlap and therefore to increase the OFM absorption signal.[31]

### 3. Modeling

In order to better understand and describe more quantitatively the broadening mechanisms associated with the specific environment of the MBE reactor, we built a model, adapted from that reported in [3], describing the forward and backward Ba absorption profiles. These Doppler-broadened profiles result from the dispersion of the projection of the atom velocities along the optical axis, which itself depends on the geometry of the system (atomic beam divergence, laser beam incidence) and on the angular distribution of atomic concentration noted $P_{\theta}(\theta)$.

The experimental system is represented in the MBE coordinate system ($\vec{x}_{MBE}, \vec{y}_{MBE}, \vec{z}_{MBE}$) in Fig. 6(a). $\vec{d}$ represents the direction of the atomic beam arising from the center of the effusion cell O and pointing toward the center of the sample. $\vec{d}$ is usually specified in the MBE coordinate system by the angles $\varphi$ and $\chi$ but for our model it is more convenient to express it as a function of the angle $\beta$ in the plane containing $\vec{z}_{MBE}$ and $\vec{d}$, noted (P). The projection of the atomic beam in (P) is shown in Fig. 6(b) in the ($\vec{x}, \vec{y}, \vec{z}$) orthonormal coordinate system. Simple geometrical considerations allow to relate the angle $\beta$ to $\varphi$ and $\chi$:

$$ \cos(\beta) = -\frac{d_{y_{MBE}}}{|d_{y_{MBE}}|} \times \frac{\sin(\chi)\sin(\varphi) + \frac{(\cos(\chi))^2}{\sin(\chi)\sin(\varphi)}}{\sqrt{1 + \left(\frac{\cos(\chi)}{\sin(\chi)\sin(\varphi)}\right)^2}}, \quad (1) $$

with

$$ \frac{d_{y_{MBE}}}{|d_{y_{MBE}}|} = \begin{cases} -1 & \text{if } 0 \le \varphi \le \pi \\ 1 & \text{if } -\pi \le \varphi \le 0 \end{cases}. \quad (d_{y_{MBE}} \text{ is the y-component of } \vec{d}) $$
<figure><img src="image_7.png" /><figcaption>Fig. 6. Schematic representing the laser beam (transparent red) and atomic beam (blue) in the MBE coordinate system (a) and in the plane containing both the laser beam and the atomic beam, noted (P) (b). The green vector $\vec{d}$ designates the cell axis, running from the center of the Ba cell to the center of the sample. $\varphi$ is the angle between $\vec{x}_{MBE}$ and $\vec{d}$ projected in the $(\vec{x}_{MBE}, \vec{y}_{MBE})$ plane. $\chi$ is the angle between $\vec{z}_{MBE}$ and $\vec{d}$. $\beta$ is the angle defined so that the angle formed by the laser beam axis and the cell axis in (P) reads 90 - $\beta$. The pink vector $\vec{r}$ is a direction in the atomic beam forming an angle $\theta$ with $\vec{z}$ emanating from an infinitesimal portion of the Ba cell surface $dS_{cell}$. $dU$ is the solid angle element subtended by $dS$ at $O$. $\vec{r}$ intersects the laser beam at point I with coordinates $(x, z)$.</figcaption></figure>

In our reactor, for the Ba Cell, $\varphi \approx 113^{\circ}$ and $\chi \approx 40^{\circ}$, leading to $\beta \approx 15^{\circ}$.

The atoms are emitted by the surface of the Ba cell $S_{cell}$. Let $\vec{r}$ be a direction in the atomic beam forming an angle $\theta$ with $\vec{z}$ emanating from an infinitesimal portion $dS_{cell}$ of this surface centered around $O$. $\vec{r}$ intersects the laser beam at point I (coordinates $(x, z)$). $dS$ designates an elemental surface parallel to $dS_{cell}$ at I. The solid angle element $dU$ subtended by $dS$ at O reads:

$$dU = \frac{dS \cdot \cos(\theta)}{r^2}, \qquad (2)$$

where $r = \|\vec{r}\|$. The number of atoms $d^2N$ evaporating from $dS_{cell}$ during an elemental duration $dt$ in $dU$ reads

$$d^2N(\theta) = Kn_{cell}dS_{cell} \frac{dU}{4\pi} P_{\theta}(\theta)\bar{v}dt, \qquad (3)$$

where $n_{cell}$ (units $m^{-3}$) is the atomic concentration in the vapor in cell above the evaporant, $\bar{v}$ is the atom velocity averaged over the entire beam, $P_{\theta}(\theta)$ is the angular distribution of atomic concentration in the atomic beam and K is a unitless constant. The atom flux through $dS$ emitted by $dS_{cell}$ (units $m^{-2} \cdot s^{-1}$) thus reads

$$\frac{d^2N(\theta)}{dtds} = Kn_{cell} \frac{dS_{cell} \cos(\theta)}{4\pi r^2} P_{\theta}(\theta)\bar{v}, \qquad (4)$$

and the atomic concentration $n$ at I (units $m^{-3}$) reads

$$n(\theta) = \frac{1}{\bar{v}} \frac{d^2N(\theta)}{dtds} = Kn_{cell} \frac{dS_{cell} \cos(\theta)}{4\pi r^2} P_{\theta}(\theta) \qquad (5)$$

Let $p_v(v, T)$ be the atom velocity distribution in the atomic beam ($v = \bar{v}$ designates the norm of the atom velocity). The probability for an atom to have a velocity between $v$ and $v + dv$ is
$P_v(v, T) = p_v(v, T)dv$, and the concentration of atoms having a velocity between $v$ and $v + dv$ at $I$ reads

$$n_v(\theta, v) = K' \frac{\cos(\theta)}{r^2} P_{\theta}(\theta) p_v(v, T) dv, \qquad (6)$$

where $K' = Kn_{cell} \frac{dS_{cell}}{4\pi}$.

As $\cos\theta = \frac{z}{\sqrt{x^2+z^2}}$, and $r^2 = x^2 + z^2$ Eq. (6) can be rewritten as

$$n_v(x, z, v) = K' \frac{z}{(x^2 + z^2)^{3/2}} P_{\theta}(x, z) p_v(v, T) dv. \qquad (7)$$

The locus of the laser beam trajectory in (P) is the straight line defined by $z(x) = z_0 + x \tan(\beta)$, with $\beta \approx 15^\circ$ (see Fig. 6(b) and its comment) and $z_0 = \|\vec{d}\| = 0.19 m$ (cell aperture – substrate distance). Thus, the concentration of atoms having a velocity between $v$ and $v + dv$ along the laser beam trajectory obeys:

$$n_v(x, v) = K' \frac{z_0 + x \tan(\beta)}{(x^2 + (z_0 + x \tan(\beta))^2)^{3/2}} P_{\theta}(x) p_v(v, T) dv, \qquad (8)$$

that is

$$n_v(x, v) = K'g(x, \beta, z_0)P_{\theta}(x)p_v(v, T)dv, \qquad (9)$$

with $g(x, \beta, z_0) = \frac{\tan\beta \cdot x + z_0}{(x^2+(\tan\beta \cdot x + z_0))^2}$. (Eq. (9)) gives the fraction of atoms at the position $x$ along a line defined by $\theta$ having a velocity between $v$ and $v + dv$.

For this class of velocity, the transition wavenumber, shifted by the Doppler effect, is expressed as:

$\sigma_c^D = \sigma_c + \frac{\vec{k}}{2\pi c} \cdot \vec{v}$, with $\sigma_c$ the transition wavenumber for atoms at rest, $\vec{k}$ the laser beam wavevector ($k = \|\vec{k}\| = 2\pi\sigma_c$) and $c = 3.10^8 m \cdot s^{-1}$ the speed of light. Expressing the scalar product as $\frac{\vec{k}}{2\pi} \cdot \vec{v} = \frac{v}{c}(\cos\beta \cdot \sin\theta + \sin\beta \cdot \cos\theta)$, $\sigma_c^D$ can be rewritten as

$$\sigma_c^D(x, v) = \sigma_c \left[ 1 + \frac{v}{c} \times \frac{x \cdot \cos \beta + (\tan \beta \cdot x + z_0) \cdot \sin \beta}{\sqrt{x^2 + (\tan \beta \cdot x + z_0)^2}} \right], \qquad (10)$$

and introducing the function $f(x, \beta, z_0) = \frac{c}{\sigma_c} \times \frac{\sqrt{x^2+(\tan\beta \cdot x+z_0)^2}}{[x \cdot \cos\beta+(\tan\beta \cdot x+z_0) \cdot \sin\beta]}$, the Doppler shifted transition wavenumber can be expressed as:

$$\sigma_c^D(x, v) = \sigma_c + \frac{v}{f(x, \beta, z_0)}. \qquad (11)$$

In case of orthogonality between the atom velocities and the laser beam, obtained for instance for $\beta = 0$ and $x = 0$, or for an arbitrary value of $\beta$ when $x = -z_0 \sin\beta \cos\beta$, $\sigma_c^D(x, v)$ is equal to $\sigma_c$, the transition wavenumber of atoms at rest, as expected.

Let $\alpha_v(\sigma, x, v)$ be the absorption coefficient of the atoms at point $I$ for this class of velocity. The spectral profile of $\alpha(\sigma, x, v)$ corresponds to the line natural broadening centered around $\sigma_c^D(x, v)$, while its area is given by $S \times n_v(x, v)$, with $S$ the transition linestrength:

$$\alpha_v(\sigma, x, v) = S \times n_v(x, v) \times \frac{\Gamma_N}{2\pi} \cdot \frac{1}{\left(\frac{\Gamma_N}{2}\right)^2 + (\sigma - \sigma_c^D(x, v))^2}, \qquad (12)$$

$n_v(x, v)$ and $\sigma_c^D(x, v)$ are given by (Eq. (9)) and (Eq. (11)), respectively, and $\Gamma_N = \frac{1}{c\tau}$ where $\tau$ is the lifetime of the upper level of the transition.
The absorption coefficient of all atoms at point I, $\alpha(\sigma, x)$, obtained by summing the contributions of all velocity classes:

$$ \alpha(\sigma, x) = \int_{0}^{+\infty} \alpha_{v}(\sigma, x, v)dv $$

which leads to:

$$ \alpha(\sigma, x) = \frac{\Gamma_N}{2\pi} SK'g(x, \beta, z_0)P_{\theta}(x) \int_{0}^{+\infty} \frac{p_v(v, T)dv}{\left(\frac{\Gamma_N}{2}\right)^2 + \left(\sigma - \sigma_c - \frac{v}{f(x, \beta, z_0)}\right)^2} \quad (13) $$

Lastly, with $\Delta = \sigma_c^D(x, v) - \sigma_c = \frac{v}{f(x, \beta, z_0)}$ the Doppler shift, a change of variable in (Eq. (13)) leads to

$$ \alpha(\sigma, x) = \frac{\Gamma_N}{2\pi} SK'g(x, \beta, z_0)P_{\theta}(x)f(x, \beta, z_0) \int_{0}^{f(x, \beta, z_0)/|f(x, \beta, z_0)|} \frac{p_v(f(x, \beta, z_0) \cdot \Delta, T)d\Delta}{\left(\frac{\Gamma_N}{2}\right)^2 + (\sigma - \sigma_c - \Delta)^2} \quad (14) $$

The absorbance, which is also the measured signal, is then simply given by integration over the optical path length which results in:

$$ A(\sigma) = \frac{\Gamma_N}{2\pi} SK' \int_{-\infty}^{\infty} \int_{0}^{f(x, \beta, z_0)/|f(x, \beta, z_0)|} g(x, \beta, z_0) P_{\theta}(x) f(x, \beta, z_0) \frac{p_v(f(x, \beta, z_0) \cdot \Delta, T)}{\left(\frac{\Gamma_N}{2}\right)^2 + (\sigma - \sigma_c - \Delta)^2} d\Delta \frac{dx}{\cos \beta} \quad (15) $$

While this relationship has been derived for the forward beam, the backward absorbance is simply obtained by the change of variables $(x, \beta) \rightarrow (-x, -\beta)$.

Equation (15) was used to fit the experimental absorption profiles with improved SNR displayed in Fig. 5(c). Both forward and backward profiles were fitted simultaneously to improve the fit consistency. For this purpose, a Maxwell-Boltzmann velocity distribution was chosen for $p_v(v, T)$:

$$ p_v(v, T) = \sqrt{\frac{2}{\pi}} \left(\frac{m}{kT}\right)^{3/2} v^2 \exp\left(-\frac{mv^2}{2kT}\right), \quad (16) $$

where $m$ is the atomic mass, $T$ the atomic beam temperature and $k$ the Boltzmann constant. Such a velocity distribution is well suited to describe the atomic beam emanating from open effusion cells as that used for these experiments (and commonly use in MBE reactors) where the crucible aperture diameter $d_c$ is not negligible with respect to the crucible length $l_c$. In contrast, for ideal Knudsen cells where $d_c$ is negligibly small, the atomic vapor is at thermodynamic equilibrium inside the cell body and the velocity distribution is described by a $v^3$ function of the form $2(\frac{m}{kT})^2 v^3 \exp(-\frac{mv^2}{2kT})$ [3]. A function describing the angular distribution of atomic concentration in the atomic beam, $P_{\theta}(\theta)$, must also be chosen. In ideal Knudsen cells where atoms effuse from the cell through an orifice of negligibly small diameter, and where the orifice walls have negligibly small thickness so that effusing atoms cannot stick on them, $P_{\theta}(\theta)$ is given by the cosine law of effusion [24]:

$$ P_{\theta}(\theta) \propto \cos(\theta). \quad (17) $$

In real effusion cells as that used for this experiment, atoms stick and re-evaporate from the crucible walls before to be emitted in the chamber, which significantly collimates the atomic beam as compared to ideal Knudsen cell case [44]. This effect can be empirically described using an atomic concentration distribution of the form [45]:

$$ P_{\theta}(\theta) \propto \cos^N(\theta) \quad (18) $$

The larger $N$, the more the beam is collimated. Besides, Clausing and co-authors [46] have developed a model to account for this collimation effect. For the sake of maintaining an analytical
form for $P_{\theta}(\theta)$, they simplified the problem by describing the cell as a tube of length $l_c$ and diameter $d_c$ connecting a reservoir at constant pressure to the growth chamber where the pressure is considered to be 0. Some atoms pass in straight line from the reservoir to the reactor without touching the tube walls. The others stick on the walls and have an equal probability of being re-emitted in any direction inside the tube, which basically consists in considering a cosine law distribution of atoms being emitted from every single surface element of the tube. In such conditions, they derived an analytical form for $P_{\theta}(\theta)$ for short tubes ($l_c \le 2d_c$) with $p = \frac{l_c}{d_c} \tan(\theta)$, the flux of molecules entering the chamber without touching the tube walls reads

$$ \Gamma_0 \propto \begin{cases} 0 & \text{if } p>1 \\ Q \left[ 1 - \frac{2}{\pi} \left( \arcsin(p) + p\sqrt{1-p^2} \right) \right] \cos(\theta) & \text{if } p<1 \end{cases} $$

and the flux of molecules entering the chamber after re-emission from the tube wall reads

$$ \Gamma_W \propto \begin{cases} Q \left[ \alpha + \frac{4(1-2\alpha)}{3\pi p} \right] \cos(\theta) & \text{if } p>1 \\ Q \left[ \frac{2\alpha}{\pi} \left( \arcsin(p) + p\sqrt{1-p^2} \right) + \frac{4(1-2\alpha)}{3\pi} \frac{1-(1-p^2)^{\frac{3}{2}}}{p} \right] \cos(\theta) & \text{if } p<1 \end{cases}, $$

with

$$ \alpha = \frac{\sqrt{l_c^2 + d_c^2} - l_c}{d_c + d_c / \sqrt{l_c^2 + d_c^2}}, $$

and $Q$ a constant. $P_{\theta}(\theta)$ then reads:

$$ P_{\theta}(\theta) \propto \Gamma_0 + \Gamma_W \quad (19) $$

Dayton and co-authors adapted this model for the cases where $l_c > 2d_c$ [47]. w

$$ \alpha = \frac{u\sqrt{u^2 + 1} - v\sqrt{v^2 + 1} + v^2 - u^2}{(u(2v^2 + 1) - v)\sqrt{v^2 + 1} + (v(2u^2 + 1) - u)\sqrt{u^2 + 1}}, $$

with $u = \frac{l_c}{d_c} - v$ and $v = \frac{l_c \sqrt{7}}{3l_c + d_c \sqrt{7}}$.

We fitted Eq. (15) to the experimental absorption profiles considering these three functions (cosN($\theta$), Clausing and Dayton) for $P_{\theta}(\theta)$. For the Clausing and Dayton functions, the dimensions of the Ba effusion cell, $d_c$ and $l_c$, were fitted by considering the actual dimensions of the cell as guess values, namely $d_c = 22.5$ mm and $l_c = 88$ mm.

In addition, we also considered a pseudo-voigt (PV) form combining a Lorentzian function of width $w_L$ to a Gaussian function of width $w_G$:

$$ P_{\theta}(\theta) = \mu \left( \frac{2}{\pi} \frac{w_L}{4\theta^2 + w_L^2} \right) + (1 - \mu) \left( \frac{\sqrt{4\ln(2)}}{w_G \sqrt{\pi}} \exp \left( -\frac{4\ln(2)}{w_G^2} \theta^2 \right) \right), \quad (20) $$

with $0 \le \mu \le 1$.

The contribution of the isotope to the absorbance, noted $A_{iso}(\sigma)$, was described using Eq. (15), by replacing $\sigma_c$ by $\sigma_c + s_{iso}$, with $s_{iso} = 0.66 \text{ GHz} (0.022 \text{ cm}^{-1})$ the isotope shift estimated from our data. The isotope transition line strength was considered equal to that of the $^{138}\text{Ba}$ transition. Designating the ratio between the isotope concentration and the $^{138}\text{Ba}$ concentration as $r_{iso}$, the total absorbance reads:

$$ A_{tot}(\sigma) = A(\sigma, \sigma_c) + r_{iso} \times A(\sigma, \sigma_c + s_{iso}). \quad (21) $$

In the end, we considered a shift along the frequency axis (noted $\sigma_{shift}$ in wavenumber) to match the calculated and experimental absorbances, the latter being measured across a relative frequency scale.
<figure><img src="image_8.png" /><figcaption>**Fig. 7.** Forward (blue) and backward (red) experimental absorbance spectra, as obtained in Fig. 5(c). Both curves are fitted (black) with the theoretical model, using the four different atomic concentration distributions: (a) Clausing, (b) Dayton, (c) $\cos^N(\theta)$ and (d) Pseudo-Voigt. Residuals are displayed in top panels.</figcaption></figure>

**4. Results and discussion**

The results of the fits obtained for a Ba cell temperature of 570°C using the four distributions of atomic concentrations are summarized in Fig. 7.

The parameter values extracted from fits are gathered in Table 1, together with their expected values when applicable.

$z_0$ (distance between the Ba load surface cell and the laser beam/atomic beam crossing point) can be estimated from the reactor dimension and is expected to be comprised between 190 and 280 mm, depending on the laser beam orientation and the effusion cell filling rate. $\beta$ has a value close to 15°, as mentioned above. Both $z_0$ and $\beta$ were refined by fitting as their value strongly impact the absorption profile and cannot be precisely determined from the reactor geometry.
<table><caption>**Table 1.** Expected parameter and fitted values. The fitted values deviating significantly from the expected values are shown in red</caption><thead><tr><th colspan="2">Parameters</th><th colspan="4">Fitted values</th></tr><tr><th>Name</th><th>Expected value</th><th>Clausing</th><th>Dayton</th><th>cos<sup>N</sup>(θ)</th><th>Pseudo-Voigt</th></tr></thead><tbody><tr><td>z<sub>0</sub> [mm]</td><td>190-280</td><td>400</td><td>840</td><td>280</td><td>270</td></tr><tr><td>β [°]</td><td>~15</td><td>20.2</td><td>18.4</td><td>15.6</td><td>16.2</td></tr><tr><td>T [°C]</td><td>~570</td><td>214.3</td><td>320.9</td><td>559.1</td><td>555.8</td></tr><tr><td>r<sub>iso</sub></td><td>~0.1</td><td>0.1</td><td>0.1</td><td>0.1</td><td>0.09</td></tr><tr><td>σ<sub>shift</sub> [cm<sup>-1</sup>]</td><td></td><td>-0.055</td><td>-0.057</td><td>-0.049</td><td>-0.053</td></tr><tr><td>N</td><td></td><td></td><td></td><td>18.9</td><td></td></tr><tr><td>d<sub>c</sub> [mm]</td><td>~22.5</td><td>27.2</td><td>26.4</td><td></td><td></td></tr><tr><td>l<sub>c</sub> [mm]</td><td>&lt; 88</td><td>53.6</td><td>60.6</td><td></td><td></td></tr><tr><td>μ</td><td></td><td></td><td></td><td></td><td>0</td></tr><tr><td>w<sub>L</sub> [°]</td><td></td><td></td><td></td><td></td><td>N.A.</td></tr><tr><td>w<sub>G</sub> [°]</td><td></td><td></td><td></td><td></td><td>32.1</td></tr><tr><td>χ<sup>2</sup></td><td></td><td>0.15</td><td>0.15</td><td>0.14</td><td>0.16</td></tr><tr><td>σ<sub>c</sub> [cm<sup>-1</sup>]</td><td>24198</td><td></td><td></td><td></td><td></td></tr><tr><td>Γ<sub>N</sub> [cm<sup>-1</sup>]</td><td>0.0019</td><td></td><td></td><td></td><td></td></tr><tr><td>S [m.at<sup>-1</sup>]</td><td>9.93 · 10<sup>-17</sup></td><td></td><td></td><td></td><td></td></tr><tr><td>m [amu]</td><td>138</td><td></td><td></td><td></td><td></td></tr><tr><td>s<sub>iso</sub> [cm<sup>-1</sup>]</td><td>0.022</td><td></td><td></td><td></td><td></td></tr></tbody></table>

Similarly, the atomic beam temperature *T*, expected to be close to the cell temperature (570°C), was refined by fitting as it is very sensitive to the position of the thermocouple used for its measurement and may therefore slightly deviate from its nominal value. The isotope ratio *r*<sub>iso</sub> and the shift along the frequency axis *σ*<sub>shift</sub> were also fitted, as well as the parameters describing the angular distribution of atomic concentration in the atomic beam, namely *N* for the cos<sup>N</sup>(θ) distribution, the crucible diameter (*d*<sub>c</sub>) and length (*l*<sub>c</sub>) for the Clausing and Dayton distributions, and the Lorentzian fraction (*μ*), the Lorentzian width (*w*<sub>L</sub>) and the Gaussian width (*w*<sub>G</sub>) for the pseudo-voigt distribution.

The absorption profiles calculated using our model are in excellent agreement with the experimental profiles, as testified by the low χ² values associated to the fits (Table 1). The four atomic concentration distributions lead to equivalently low χ² values. However, the Clausing and Dayton distributions lead to inconsistently high values for *z*<sub>0</sub>, incompatible with our reactor geometry, and to inconsistent values for the beam temperature *T*, well below the nominal cell temperature of 570°C (red figures in Table 1). They therefore do not satisfactorily describe the angular concentration profile in the atomic beam. In contrast, the fits performed using the cos<sup>N</sup>(θ) and pseudo-voigt distributions lead to consistent values for all fitted parameters. In particular, the fitted *z*<sub>0</sub> values are consistent with that expected from the reactor geometry, and the fitted atomic beam temperatures are close to the nominal cell temperature of 570°C, which suggests that these two distributions realistically describe the concentration profile in the atomic beam. It may also be noted that the fits lead to *r*<sub>iso</sub> values of the order of 0.1, very close to the relative abundance of the main Ba isotopes [48].

Interestingly, the two different analytical forms used for these two distributions lead to quasi identical angular concentration profiles, and to very close values for all the fitted parameters. This supports the conclusion that this profile, extracted from the fit, reliably describes the angular distribution of atomic concentration in the atomic beam. Furthermore, this distribution is
significantly narrower than that expected for an ideal Knudsen cell (cosine law of effusion, grey dashed line in Fig. 8) as the atomic beam is strongly collimated by the cylindrical crucible used for Ba effusion. In the end, our result also show that the Clausing and Dayton distributions do not describe the angular concentration profile in the atomic beam in our experimental conditions. In particular, the Clausing and Dayton distributions are less collimated and present wings at high divergence angles, due to the re-evaporation of atoms from the cell walls, which are absent in the $cos(\theta)^N$ and pseudo-voigt distributions that best describe our experimental results. From this point of view, it is interesting to note that for the PV distribution, the best fit is obtained for a $\mu$ value equal to 0. The Lorentzian contribution, which gives the PV distribution its broadening at high divergence angles, is therefore zero, and the distribution is a pure Gaussian, better describing the collimated nature of the profile. Clearly, the wall re-evaporation mechanisms of the atoms in the Ba cell are less efficient than those described in the Clausing and Dayton models. The Clausing distribution is only valid when $l_c \le 2d_c$ [46,49], condition that is not verified in our experiment for which a cylindrical crucible of length $l_c = 88$ mm and diameter $d_c = 22.5$ mm was used. The Dayton distribution suitably describes cylindrical crucible geometry with $l_c>2d_c$. However, Dayton's model supposes that any impinging atom on the crucible walls reflects according to the cosine law from its impinging point, without diffusing. In real effusion cells however, atoms impinging on the walls diffuse along the walls before to be re-emitted. Such effects, highly dependent on the wall temperature and on the atom/wall chemistry, have been shown to significantly affect the atomic concentration distribution at the cell output [50]. The fact that they are not taken into account in Clausing and Dayton models may explain why the latter fail to describe our experimental results.

<figure><img src="image_9.png" /><figcaption>**Fig. 8.** Angular distribution of atomic concentration in the atomic beam extracted from the fit of the experimental absorption profiles using different concentration profiles with the from Table 1. The $cos(\theta)$ distribution expected for an ideal Knudsen cell is also represented for comparison.</figcaption></figure>

**5. Conclusion**

As a conclusion, the results reported here highlight the influence of the reactor geometry on the spectral shape of the absorption of atomic beams produced in a MBE environment. The latter is
measured for the 6s² ¹S-5d6p ³D⁰ Ba absorption line using an optimized TDLAS setup, leading to excellent measurement accuracy (SNRs up to 100) despite the noisy environment of the MBE reactor. The model we propose realistically describes the experimental results by taking into account these geometrical effects as well as the angular distribution of atomic concentration in the atomic beam. It shows that the latter is neither accurately described by the Dayton model, nor by the Clausing model. This model is also a useful tool to design OFM sensors and better understand their response. In particular, our results show that the Ba absorption line has a Doppler width of about 500 MHz, significantly smaller than the typical width of HCL lines, in the 1.5-3 GHz. This difference affects OFM sensors due to imperfect overlap between both lines. Our results also show that the forward and backward absorption spectra are spectrally shifted from each other leading to an improved effective overlap between the HCL emission line and the absorption line in two-passes OFM sensors. More generally, our results show that the spectral overlap between the HCL and the absorption line not only depends on the atom velocity in the beam, as explained in [31], but also on the geometry of the reactor and of the setup, due to Doppler shift. Our model can be adapted to account for the hyperfine structure of elements with non-zero nuclear spin, and can therefore be used to estimate the absorption profile for any element in any reactor geometry.

**Funding.** Agence Nationale de la Recherche (ANR-21-CE24-0003).

**Acknowledgments.** The authors thank the Nanolyon platform for their support in the MBE experimental work.

**Disclosures.** The authors declare no conflicts of interest.

**Data availability.** Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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