## Introduction

The Beer-Lambert exponential transmission law1,2 describing attenuation of monochromatic light by the homogeneous, not very dense medium is well known for almost three centuries. Despite developing newer, more advanced transmittance models, today it still applies to quantitative spectroscopy3 and rarefied gases, among others. All this models relies on an assumption of attenuating particles locality. However, an increasing number of experiments4,5 convince us that the underlying theory of Quantum mechanics is not a local realistic theory6,7. There is one more assumption in most of “classic” transmittance models: a light detector is a macroscopic apparatus. Quantum mechanics is considered to be one of the most fundamental theories so it is necessary to check whether these two assumptions limit scope of applicability of classic models.

Quantum spreading is an effect that involves spatial smearing of the $$\Psi$$ wave function over time. It leads to the spreading of the $$|\Psi |^2$$ density of the probability of any reaction (quantum measurement) of a physical object described by such a function. It is derived from solving the Schrödinger equation for a free particle8. We applied this solution to each gas particle independently during its free time between successive collisions. It is a kind of “smeared gas”. It leads, together with the assumption of non-locality, to a new model9 of electromagnetic transmittance of thin gases. One of the predictions of this model is that the measured optical transmittance depends, among others, on the size of the detector used and the duration of the particles mean free time. The classical, “local” approach to transmittance, the Beer-Lambert law included, does not predict any of such dependencies.

Currently, one of the most popular methods of quantitative testing the transmittance of thin gases is Tunable Diode Laser Absorption Spectroscopy (TDLAS)10. Our setup is a slightly modified version of such a system. The transmittance is measured simultaneously using a pair of detectors with different effective active cross-sections. One laser beam passes through the gas and eventually it’s split towards two detectors with a beam-splitter. There is a big vacuum chamber used to provide conditions for a significant spreading of the gas particles wave functions to occur. The model applies to any type of gas. We chose to test the water vapor.

The objective of the experiment is to examine the qualitative prediction on whether the transmittance measured with a smaller detector is greater than the transmittance measured with a larger detector. Results of the experiment confirm this with $$>5\sigma$$ statistical significance.

The experimental setup is described in the next section. The way data is acquired and processed is covered in the following sections. Results are discussed in the 5th section. Conclusions are presented at the end of the paper. The measurement uncertainty is analyzed in the appendix next to the online data guide.

## Experiment

### Setup

The schematic diagram is shown in Fig. 1. The tested gas, along with the optical setup and the detectors, is placed inside a bell-type vertical cylindrical vacuum chamber (ø $$\approx{58}\,{\hbox {cm}}$$, $$h\approx{110}\,{\hbox {cm}}$$) in room temperature. The working pressure is approx. $$10^{-3}-10^{-2}$$ mbar. A DFB SM-pigtailed laser diode (LD-PD Inc., PL-DFB-1368-A-A81-SA) operating at $$\sim1368.60\,{\hbox {nm}}$$ with 2 MHz line width is used to scan 101313–00021211 water absorption line. The DFB CW output power is about 1 mW at $${27}\,{^\circ }$$C. The laser diode is equipped with an internal InGaAs photodiode. The DFB diode is controlled by an integrated current and temperature controller (Thorlabs, CLD1015) placed outside the chamber. The temperature setting of the diode is set to $${27}\,{^\circ }$$C. The laser beam is fed into the chamber using single-mode optical fibers (SMF-28E and SMG652.D) using vacuum feedthrough (SQS, KF40 SM). The beam is collimated with adjustable ECO-550 lenses (Thorlabs, PAF2-2C). The vertical multi-pass setup is made up of two 3” dielectric concave mirrors f = $${500}\,{\hbox {mm}}$$ (Thorlabs, CM750-500-E04-SP) held vertically by metal rods about $${76.7}\,{\hbox {cm}}$$ apart from each other. Two flat silver mirrors (Thorlabs, MRA03-P01, MRA10-P01) are used to direct laser beam from the collimator to/from the multipass cell and towards a 92/8 pellicle beamsplitter (Thorlabs, CM1-BP108). The first flat mirror that directs the laser beam up to the multipass cell is approx. $${8}\,{\hbox {cm}}$$ from the collimator. The beamsplitter is approx. $${20}\,{\hbox {cm}}$$ away from this mirror. Detector 1 is approx. $${5.2}\,{\hbox {cm}}$$ from the beamsplitter, Detector 2 is $${6.7}\,{\hbox {cm}}$$ away. The total light path length is approx. $${61.4}\,{\hbox {m}}$$ long and it is $$\sim {1.5}\,{\hbox {cm}}$$ shorter towards Detector 1. The multipass cell axis is in line with the chamber vertical axis to assure equally long distance from the chamber walls, approx. 25 cm. All optical components are installed on a small breadboard standing on the chamber floor, see Fig. 2.

Transmittance is measured in parallel by a pair of ø200 $$\upmu \hbox {m}$$ InGaAs photodiodes (Hamamatsu G11193-02R). A circular aperture stop to reduce detector’s effective active area is inserted in front of the detector ($$Ch_1$$) placed in line with the beamsplitter incoming beam, see Table 1. The aperture is installed $$\sim {1}\,{\hbox {mm}}$$ away from the photosensitive surface. The second detector ($$Ch_2$$), without an aperture, gives a reference transmittance reading. Both detectors and the aperture stop are installed in their own X–Y translation units. The $$\sim \,{92}\%$$ of the beam is directed towards the first detector.

Both photodiodes operate in a photovoltaic mode12 with no bias voltage. The photodiode current is amplified13 by a custom-built current-to-voltage amplifier using the low current op-amp (TI, LMC662), followed by a low-pass $${10}\,\hbox {Hz}$$ RC filter. The amplifier gain is controlled by a feedback loop resistor. The $$Ch_1$$ resistance is adjusted for each aperture individually to keep output voltage within a range of a few dozen mV, see Table 1. The $$Ch_2$$ resistance is constant and equal to 2k$$\Omega$$. Gains are chosen to obtain both characteristics curves as linear as possible within the range used for actual measurement.

Vacuum is evacuated with a scroll pump and a turbo molecular pump. Two independent vacuum meters are used (Leybold, Ceravac CTR 100N, Ionivac ITR90).

All voltage data is acquired by the DAQ unit (LabJack, U6). The digital data, from the DAQ unit and the laser driver, is retrieved via a USB by a PC computer running LabView. A custom built LabView GUI is used to automate the measurement procedure. The LabView text files data is extracted periodically into the SQL database from where it is retrieved for statistical analyses performed by the Mathematica 12 package.

In this paper $$Ch_1$$ and $$Ch_2$$ may denote either a respective photodiode or one of measurement channels or a voltage read from one of channels.

### Gas and spectral line selection

We selected water vapor due to its large cross-section in the near infrared. The HITRAN14 and MARVEL15 data was referenced. The large cross-section allows for testing transmittance at low pressures with shorter path of light. Low pressure and a narrow spectral range of the laser allow for isolating a single absorption line with sufficient resolution.

For our setup the expected mean free path of water particles for the pressure of $$\sim {10^{-3}}\,{\hbox {mbar}}$$ is longer than $${37}\,{\hbox {cm}}$$. With approximate mean speed in the room temperature of $${600}\,{\hbox {ms}^{-1}}$$ it gives approx. $${0.6}\,{\hbox {ms}}$$ particle mean free time. Plugging this time to the Schrödinger equation free particle solution we calculate the particle wave packet standard deviation to be $$\sim {14.6}\,\upmu \hbox {m}$$. This is a conservative calculation using ideal gas laws.

The chamber size is another restriction on the particle mean free path. There should be enough distance between walls and the measuring laser beam. This is why a chamber diameter of $${58}\,{\hbox {cm}}$$ was chosen.

### Stages of the experiment

The experiment is carried out in successive, several days long runs, see Table 2. There is a different aperture stop installed during each run. A run consists of many consecutive either $$\sim {30}\,{\hbox {min}}$$ or $$\sim {10}\,{\hbox {min}}$$ long cycles.

Replacement of an aperture requires opening of the vacuum chamber. After the aperture installation, the $$Ch_1$$ gain is adjusted by soldering in the appropriate feedback loop resistor. In case of two runs (R-8 and R-10), the aperture stop was removed in order to perform a comparative measurement for two detectors equal (according to manufacturer) in size.

After the chamber has been closed, vacuum is build to $$\sim {10^{-5}}\,{\hbox {mbar}}$$, which takes roughly 1–2 days. In some cases, the decision was taken to extend this period to 3–5 days. Water is sucked in from an external tank due to the pressure difference. Dosing is controlled using a needle valve adjusted by a manual micrometer screw. Water vapor is produced as a result of spontaneous boiling of water at low pressure.

Water vapor tends to settle on the inside walls of the chamber and setup components16. This can be seen in the spontaneous pressure drop, starting immediately after the feed valve has been closed. To obtain the working pressure in the order of $${10^{-3}}-{10^{-2}}\,{\hbox {mbar}}$$, the valve is kept open for a longer period of time (e.g., an hour), while maintaining higher pressure (e.g., $${2\times 10^{-2}}\,{\hbox {mbar}}$$), see pressure plots in Fig. 9.

The pressure measurement below $${10^{-3}}\,{\hbox {mbar}}$$, made using the Ionivac vacuum gauge ($$PRS_B$$), is burdened with quite a large error, in the order of $$\pm {15}\%$$. However, this range is used solely to build vacuum prior to feeding the water. The total working pressure, i.e. above $${10^{-3}}\,{\hbox {mbar}}$$, is measured, with the accuracy of $$\pm {0.5}\%$$, using the Ceravac vacuum gauge ($$PRS_A$$). The partial pressure of the water ($$PRS_{H_2O}$$) may be determined by converting the classical transmittance measured with $$Ch_2$$, see Eq. 6. The partial pressure can be determined in our setup within the range of approx. $${1\times 10^{-3}}-{3\times 10^{-2}}\,{\hbox {mbar}}$$.

To purify the chamber atmosphere vacuum may be build and water may be fed many times during a single run without opening the chamber. These operations are visible as discontinuities on pressure plots in Fig. 9.

## Data acquisition

Within a single run the measurement is made in cycles composed of following steps:

1. 1.

measuring detectors characteristic curves,

2. 2.

tuning in the wavelength of the laser light to the chosen absorption line,

3. 3.

measuring both transmittances,

4. 4.

measuring detectors characteristic curves again (with the laser current step reversed).

The 4th step is the 1st step of the next cycle. Fig. 3 presents sample readings of the $$Ch_2$$ detector during two successive cycles. Cycles last either $${30}\,{\hbox {min}}$$ or (in the 2nd part of the experiment) $${10}\,{\hbox {min}}$$. All cyclic measurements are fully automated with a LabView app. Each cycle has its own, time-based identifier assigned e.g., 20200314-0531. The identifier is referred during data post processing and data presentation.

### Detector characteristics acquisition (step 1)

The setup is susceptible to various types of drifts: laser tuning, changes in light polarization in optical fibers (affecting the detector readings), pressure and mechanical changes. They cause, among others, a change of the detector characteristics. Due to the adopted method of determining the transmittance quotient, acquiring the appropriate voltage versus light power characteristic curves is of key importance: $$P_1(Ch_1)$$ and $$P_2(Ch_2)$$. Although they are called for convenience “detectors characteristics”, they are not just photodiodes characteristics. They cover all light & signal processing steps in turn: the fibers & mirrors light attenuation, the aperture stop ($$Ch_2$$ only), the photodiodes light-to-current conversion, the current-to-voltage conversion & amplification and A/D conversion.

The characteristics are determined by changing the DFB diode current $$I_{LD}$$. Based on the fact that the DFB diode output power is proportional to the current set, the detector response is determined as a function of the laser power. The range of current changes is automatically selected based on the last reading (from the previous cycle) of the transmittance in the absorption line. It is assumed that the transmittance in the current cycle will be similar to the previous one, i.e. that the partial pressure of the water vapor does not change significantly.

The span of the $$I_{LD}$$ current changes is arbitrarily set as either $${\Delta }I_{LD}=\pm \,{12}\,{\hbox {mA}}$$ for $${30}\,{\hbox {min}}$$ long cycles or $${\Delta }I_{LD}=\pm \,{9}\,{\hbox {mA}}$$ for the shorter ones. DFB controller’s minimum current step is $${0.05}\,{\hbox {mA}}$$. It is known that a change of the DFB diode current also causes a change of the wavelength. In case of the diode used in the experiment $${\Delta }{\lambda }/{\Delta }I_{LD}\approx {78}\,{\hbox {pm}\,\hbox {mA}^{-1}}$$, which for (current span of) $${24}\,{\hbox {mA}}$$ leads to a change of the wavelength by $$\sim {1.8}\,{\hbox {nm}}$$. We chosen span and range to avoid overlapping with other strong absorption lines while characteristics acquisition.

Such a change in wavelength may also cause a slight change in detector sensitivity (up to $${1}\%$$ according to the manufacturer). It is assumed, however, that due to the use of two detectors from the same series, the relative change in sensitivity between the detectors is the same. The same offset cancels possible measurement error because it appears both in the numerator and the denominator of the formula used to determine the transmittance quotient, see Eq. (3).

It is required that the actual laser temperature is within the $$\pm {0.003}\%$$ range from the set point of $${27}\,{^\circ }\hbox {C}$$ for each measurement. The scan starts $${250}\,{\hbox {ms}}$$ after reaching required temperature range. Each recorded measurement within a scan is the arithmetic mean of 30 consecutive single readings ($$Ch_1$$, $$Ch_2$$, $$P_{LD}$$) made every $${11}\,{\hbox {ms}}$$. Despite allowing a fairly narrow range of stabilized temperature, the current change direction (up/down) is switched in each cycle. It’s either +$${0.05}\,{\hbox {mA}}$$ or $${-0.05}\,{\hbox {mA}}$$. A constant increase (or a decrease, accordingly) of the laser current $$I_{LD}$$ makes the laser diode working temperature always a little higher (or lower, accordingly) on average. This is because the temperature compensation circuit (Peltier TEC, PID) keeps up with the current changes quite slowly. To compensate this effect the measurements obtained both at the beginning and at the end of the cycle (steps 1 & 4) are used together to calculate the characteristic curve for the given cycle. This way up to $$2{\times }2{\times }12/0.05=960$$ (or later $$2{\times }2{\times }9/0.05=720$$) points are available for approximate characteristics models.

The scan range may be automatically limited from the bottom. For higher pressures (in the order of $${10^{-2}}\,{\hbox {mbar}}$$) the absorption line is so deep that only a few percent of the light reaches the detectors. In such case the characteristic is obtained close to the lower limit of the laser action and it is impossible to measure some of the distant points in the $${\Delta }I_{LD}$$ negative range. The measurement of the characteristics is then limited so that the $$I_{LD}$$ current fed is large enough (min. $${3}\,{\hbox {mA}}$$) to excite the laser action.

The scan range is limited from the top as well. Laser current $$I_{LD}$$ can’t exceed $${52}\,{\hbox {mA}}$$, so that, by accident, no measurement of the water absorption line is made.

The characteristic acquisition takes approx. 6 minutes. Most of this time is spent waiting for temperature stabilization of the laser.

### Tuning in the wavelength (step 2)

After the characteristics has been measured, a fixed value of the laser current $$I_{LD}$$ is set to get the wavelength as close as possible to the absorption line maximum. Because of drifts of the diode and the controller it’s necessary to adjust the laser current $$I_{LD}$$. We check, for 3 distinct settings, which transmittance in the $$Ch_2$$ detector is the lowest: one that is identical as in the previous cycle or one that is different by either plus or minus $${0.05}\,{\hbox {mA}}$$. Checking starts 45 seconds after the measurements of the characteristics had been completed, providing time to properly relax temperature of the laser to the range of $$\pm {0.003}\%$$. This way, actual wavelength is kept as close as possible to the absorption maximum during the entire experiment.

### The transmittances measurement (step 3)

Having set the fixed laser current and temperature setpoints near the absorption line maximum, the transmittances measurement is carried out in the continuous mode. We don’t use any wavelength modulation techniques. Single measurements are recorded twice a second. A single measurement includes following parameters read synchronously: both detectors voltages ($$Ch_1$$, $$Ch_2$$), laser power read from the laser photodiode ($$P_{LD}$$), actual laser temperature and current ($$T_{LD}$$, $$I_{LD}$$) as well as both pressure readings ($$PRS_A$$, $$PRS_B$$). This measurement step ends half an hour (or $${10}\,{\hbox {min}}$$) from the start of the cycle.

### The 2nd characteristics acquisition (step 4)

The last step in the cycle is to acquire measurements for the characteristics again. This time, stepping the laser current in the reverse direction. It is also the first step of the next cycle.

## Data processing

The difference of the water vapor transmittances measured by both detectors is calculated as the transmittance quotient (TQ) according to the following procedure.

### Determination of the linear model of the detector characteristics

The characteristics $$P_1(Ch_1)$$ and $$P_2(Ch_2)$$ of both detectors are determined for each measurement cycle. They are approximated with a linear regression near the range of the detector voltage values observed during the given measurement cycle:

\begin{aligned} P_1(Ch_1)= & {} A_1 Ch_1+B_1 , \nonumber \\ P_2(Ch_2)= & {} A_2 Ch_2+B_2 , \end{aligned}
(1)

where $$A_{1,2}$$ and $$B_{1,2}$$ are linear model coefficients.

There are up to 960 (or 720) points ($$Ch_{1n}$$, $$P_{LDn}$$), ($$Ch_{2n}$$, $$P_{LDn}$$) available for each linear regression fitting. $$P_{LD}$$ is the power of light emitted by the laser as reported by the DFB built-in photodiode, $$Ch_{1,2}$$ is the detector’s voltage reading, the n index denotes the subsequent readings during scans. See Fig. 4.

A set of points used for actual fitting is limited to the range of voltages measured by the detector within the cycle. It gives linear approximation with a smaller error within the interesting range. Usually these are very narrow ranges of detector operation (e.g. tenths of $${\hbox {mV}}$$), resulting from noise and a small pressure difference during the cycle. Very often they are so narrow that there are too few measurements available for accurate model fitting. The range is then extended up to $${25}\%$$ of the voltage range of the characteristic being measured. This way the model is fitted using approx. 150–200 points only. Thanks to the surplus of points, the linear regression can be assessed visually in a broader context, see overview in Fig. 4. Such plots are available online for all measurement cycles.

The quality of the approximation is assessed for each cycle by independently using parameter $$R^2$$. The residuals graph may be also examined, see Fig. 5. Usually cycles where pressure fluctuations are too large (i.e. $$Max(PRS_B)-Min(PRS_B)>{5\times 10^{-3}}\,{\hbox {mbar}}$$) are skipped because there is no way to find a good enough linear approximation. Most often these are the first 1-2 cycles after feeding water into the chamber.

The measurements obtained at both ends of the given cycle are used together to determine the detectors characteristics models. They always differ based on the direction of the laser current change (“up” or “down”). This way, any potential systematic shifts of the characteristics related to the laser temperature relaxation is eliminated. The triangles direction in Figs. 4 and 5 denote measurements from two different directions of obtaining the characteristics.

We chose the parameters of the experiment (i.e. temperature and range of the laser current, absorption line, working pressure and length of the light path) in the way the characteristics are determined in a range where other water absorption lines are sufficiently weak. Therefore, water vapor do not cause any disturbances of the detector characteristics model calculations.

### Transmittance Quotient calculation

Gas optical transmittance is the ratio of the power of the light beam that has passed through the medium divided by the power of the light that has entered the medium. The characteristics of the detectors calculated above allow for calculating the transmittance for each of the channels:

\begin{aligned} TR_1= & {} P_1(Ch_1)/P_{LD} , \nonumber \\ TR_2= & {} P_2(Ch_2)/P_{LD} . \end{aligned}
(2)

All “constant” light power loses for both measurement channels are encoded within $$P_1$$ and $$P_2$$ models, respectively. These loses mainly result from mirrors reflectivity, photodiodes efficiency and beam splitting. This way the Eq. (2) are valid for determining just water vapor transmittance.

We compare transmittance values using the transmittance quotient (TQ) as defined below:

\begin{aligned} TQ=\frac{TR_1}{TR_2}= \frac{P_1(Ch_1)/P_{LD}}{P_2(Ch_2)/P_{LD}}=\frac{P_1(Ch_1)}{P_2(Ch_2)}~. \end{aligned}
(3)

Note that the quotient defined above is not a simple quotient of light power measured by both detectors. It is the quotient of transmittances measured by both detectors.

There is either transmittance difference or transmittance quotient referred in this report. Transmittance difference is expressed as a percentage so it is just the transmittance quotient minus 1. TQ values greater than 1 (transmittance difference greater than 0) mean that the transmittance measured by the smaller detector $$Ch_1$$ is greater than the transmittance measured by the classical one $$Ch_2$$. Examples of the TQ values for the individual measurements are shown in Fig. 6.

There is a possible issue with a non linearity of the DFB laser diode (current vs output power) and the DFB photodiode (incident light vs current). It was checked that this non linearity is so small in the operational range that it doesn’t significantly affect transmittance measurement. It should be taken into account for very accurate transmittance measurements. Besides we’re interested just in transmittance quotient in this experiment. This non linearity cancels out because it appears in both numerator and denominator of Eq. (3).

The actual reading (including its error) of the laser photodiode ($$P_{LD}$$) is irrelevant for the calculation of transmittance quotient TQ because it cancels out appearing in both numerator and denominator of Eq. (3).

Determination of the transmittances and their quotient is possible only if the domains of characteristics $$P_{1,2}$$ includes the appropriate range of the $$Ch_{1,2}$$ voltages. It is checked automatically for each cycle. If the appropriate characteristic range is missing the measurements made within the given cycle are skipped by the aggregating algorithm. It may happen sometimes for sudden pressure changes.

There is a small ($$\sim {1.5}\,{\hbox {cm}}$$) difference in total light path length towards detectors. Because light path towards Detector 1 is $$\sim0.024\%$$ shorter transmittance measured by Detector 1 is systematically $$\sim0.024\%$$ higher. We will see later that such a difference is negligible but it should be noted.

### Transmittance Quotient aggregation

We determine the transmittance quotient for each single measurement within a cycle. There are 2 measurements per second recorded which makes hundreds measurements per cycle. Assuming that the experimental conditions are sufficiently constant during the cycle, the average transmittance quotient $$\overline{TQ_k}$$ for the entire cycle is determined:

\begin{aligned} \overline{TQ_k}= & {} \frac{1}{M_k}\sum _{m=1}^{M_k}\frac{P_{1k}(Ch_{1m})}{P_{2k}(Ch_{2m})}\nonumber \\= & {} \frac{1}{M_k}\sum _{m=1}^{M_k}\frac{A_{1k}Ch_{1m}+B_{1k}}{A_{2k}Ch_{2m}+B_{2k}} \end{aligned}
(4)

where k-cycle index, $$M_k$$-number of the single measurements in the k-th cycle, m-measurement index within a cycle, $$A_{1,2k}$$, $$B_{1,2k}$$-linear regression factors of $${P_{1,2k}}$$ characteristics, respectively for channel 1 and 2.

Determination of the transmittance quotient $${\overline{TQ}}$$ for the entire run (for single aperture) involves calculating the arithmetic mean of the quotients measured in the successive cycles:

\begin{aligned} {\overline{TQ}}=\frac{1}{K}\sum _{k=1}^{K}\overline{TQ_k} \end{aligned}
(5)

where K-number of cycles in a run.

### Water vapor partial pressure calculation

Having determined the classical transmittance in the $$Ch_2$$ channel, the partial pressure of water vapor ($$PRS_{H_2O}$$) can be determined based on the relationship between transmittance, absorbance and pressure:

\begin{aligned} PRS_{H_2O}{\approx -0.00854}\,Log(TR_2)\,{\hbox {mbar}}~. \end{aligned}
(6)

We calculated the conversion factor depending on light path length, absorption line cross-section and temperature.

## Results and interpretation

The aggregated results of the experiment are presented in Fig. 7. The mean transmittance quotient $${\overline{TQ}}$$ is shown for each individual aperture examined. The rightmost point is the result of no aperture runs. All measured values are greater than 100%. It means that the $$Ch_1$$ (smaller) detector measured higher transmittance.

All presented quantitative results (transmittance quotient and transmittance difference) don’t take into account the systematic shift introduced by the $$\sim0.024\%$$ shorter light path towards Detector 1. This is because one may simply shift the reference base from 100% to 100.024% when analyzing if transmittance measured with the smaller detector is greater or not.

The more detailed graphs showing the values of measurements for the individual cycles along with pressure and transmittance are presented in the Appendix, see Fig. 9. Yet more data and figures are available online at www.smearedgas.org/experiment1. Among others, there are “cycle cards” prepared for all measurement cycles. They presents all essential data collected during single cycle along with parameters calculated, see Appendix for the guide. Raw data is available upon request.

### Runs with aperture stops

We observe that all runs with the pinhole installed show higher transmittance when measured by the smaller $$Ch_1$$ detector! This is exactly the effect qualitatively predicted by the smeared gas theory. For runs with bigger, namely 25-, 40-, 50-, 100- and $${150}\,\upmu \hbox {m}$$ apertures, the statistical significance is higher than $$5\sigma$$.

The maximum transmittance difference equals to $$1.23\,\pm 0.1$$ %, when the $${50}\,\upmu \hbox {m}$$ aperture is installed. For the 100- and $${150}\,\upmu \hbox {m}$$ apertures the difference is equal to $$0.52\,\pm 0.06$$ % and $$0.53\,\pm 0.07$$ % respectively. They are smaller than the $${50}\,\upmu \hbox {m}$$ aperture result. This is in line with the model’s qualitative prediction: as the detector size increases, the measured transmittance should decrease—for detectors significantly larger then the wavelength (a kind of geometric optics approximation).

This relationship is the other way around for the 6 smallest apertures, however. Furthermore, for the 3 smallest apertures the statistical significance falls down to 2–3$$\sigma$$. This is because the transmittance, according to the smeared gas model, is proportional to the definite integral of $$|\Psi |^2$$ over so-called “visibility tunnel”. The visibility tunnel is a volume where a photon amplitude doesn’t cancel out when using certain detector - according to the path integrals approach17. For detectors comparable in size to the wavelength the non-cancelling photon probability amplitudes outside the “classic” visibility tunnel effectively thicken this tunnel. This increases the volume over which the probability distributions of the smeared gas particles are integrated. As a consequence, the likelihood of observing a photon scattering event with such a small detector rises. Therefore, the measured transmittance decreases—in the extreme case down to the classical transmittance level. We see this kind of relation on the left side of the Fig. 7. The transmittance quotient is decreasing towards 100% along with decreasing $$Ch_1$$ detector aperture diameter (from 50- down to $$2\,\upmu \hbox {m}$$).

The result for the $$100\,\upmu \hbox {m}$$ aperture (with much higher average pressure) is discussed later on. The very low reading for the $$15\,\upmu \hbox {m}$$ aperture needs further analysis.

The quantitative model fitting to experimental results will be performed further down the line. Still, we observe qualitative predictions are met in the experiment.

### Control runs without aperture

During the R-8 and R-10 runs there are no apertures installed. Transmittance is measured using two naked detectors with a comparable diameter. The measured transmittance difference is $$0.12\,\pm 0.04$$ %. The transmittances are thus not perfectly identical but very close to each other.

Apart from obvious measurement errors, such difference may result, for example, from the diameter tolerance due to the workmanship of the detectors themselves, impurities etc. This tolerance, in turn, is irrelevant in case of the measurements made with aperture stops in place because detector $$Ch_1$$ is obscured by the aperture anyway. Another reason for the observed difference may be a potential variance in the sensitivity of both detectors to different wavelengths as discussed earlier. This may cause some systematic error when determining the characteristics of all measurements. However, this control runs would indicate at least the order of magnitude of such a systematic error. Even if it is $${\sim }0.12$$ %, it is still significantly smaller than the measured transmittance differences for 5 apertures larger than $${25}\,\upmu \hbox {m}$$. Consequently, it does not undermine the conclusions drawn.

### Transmittance quotient versus pressure

There are at least two phenomena influencing the transmittance quotient that should be taken into account when the pressure changes and temperature is kept constant. As the pressure increases, (1) the number of molecules increases (the transmittance drops), (2) the mean free path shortens. The experiment was not designed to examine these correlations so very little may be concluded in this field. However, there are some indications we can take a closer look at. They are in accordance with the model.

### Transmittance quotient versus number of molecules

According to the model the increasing number of molecules with the mean wave function spread unchanged (aka the mean free time constant) along with a fixed visibility tunnel should increase the transmittance quotient. We show it qualitatively on the Fig. 8. There are presented sample predictions of the model in a range of “geometric optics” approximation for two different water vapor partial pressures: $$\sim {1\times 10^{-2}}\,{\hbox {mbar}}$$ and $$\sim {6\times 10^{-3}}\,{\hbox {mbar}}$$. According to Eq. (6) they correspond to transmittance ($$TR_2$$) of 30% and 50% respectively. The transmittances for each detector $$d=1,2$$ are calculated with the following equation9:

\begin{aligned} TR_d({\bar{t}}) = \prod _{n=1}^N \left( 1-\frac{G_d()}{4} \left[ erf \left( \frac{o_n-r_{Td}}{\sqrt{2} stdev_{A_n}({\bar{t}})} \right) - erf \left( \frac{o_n+r_{Td}}{\sqrt{2} stdev_{A_n}({\bar{t}})} \right) \right] ^2 \right) ~, \end{aligned}
(7)

where $${\bar{t}}$$ denotes the molecules mean free time, N is a number of molecules in the chamber, $$o_n$$ is the n-th molecule distance from the beam. The standard deviation $$stdev_{A_n}({\bar{t}})$$ is assumed to be $$\sim {14.6}\,\upmu \hbox {m}$$ (see next paragraphs). The geometry coefficient $$G_d()$$ is equal to cross section of the 101313-00021211 water absorption line ($$\approx {7.76\times 10^{-19}}\,{\hbox {cm}^{2}}$$) divided by the area of the detector. Theoretical transmittance quotients are calculated assuming the transmittance tunnel has the shape of a $${60}\,{\hbox {m}}$$ long truncated cone. The smaller base is of the size of an aperture. The angle (an effective divergence towards the source) is $${0.008}\,{\hbox {mrad}}$$. Note, that this divergence is much smaller than the typical laser beam divergence thanks to the spherical mirrors re-focusing the beam for 80 times. The $$r_{Td}$$ is the radius of the (cylindrical) visibility tunnel, it’s different for each detector d. For simplifying calculations the conical tunnel is approximated as a cylinder, thus $$r_{Td}$$ is equal to the mean of both bases radii.

There are TQ measurements shown on the Fig. 8 for at least 4 different apertures: 25-, 40-, 50- and $${150}\,\upmu \hbox {m}$$ following the model qualitative prediction: $${\overline{TQ}}$$ is higher for the higher pressure. The statistical significance of this relation reaches as much as 3$$\sigma$$ for the 25- and $${40}\,\upmu \hbox {m}$$ apertures. Measurements for other apertures and pressures are omitted due to missing comparable data.

### Transmittance quotient versus mean free path

The measured transmittance quotient for the $${100}\,\upmu \hbox {m}$$ aperture stop is smaller than ones measured for 50- and $${150}\,\upmu \hbox {m}$$ apertures, see Fig. 7. According to the model TQ closer to 100% may indicate a shorter mean free time. Indeed for the $${100}\,\upmu \hbox {m}$$ aperture stop, most of the time the total pressure was higher than $${3\times 10^{-2}}\,{\hbox {mbar}}$$ and the water vapor partial pressure $$PRS_{H_2O}$$ was higher than $${1\times 10^{-2}}\,{\hbox {mbar}}$$. The measurements for other apertures was made at a lower total pressure, approx. in a range from $${1\times 10^{-3}}\,{\hbox {mbar}}$$ to $${3\times 10^{-2}}\,{\hbox {mbar}}$$. It seems that so high pressure eventually led to shortening the mean free path to a value smaller than the one constrained by the chamber size. This way the water molecule wave function standard deviation might drop below $${14.6}\,\upmu \hbox {m}$$ lowering the $${\overline{TQ}}$$ reading.

## Further works

First of all it is necessary to conduct the quantitative model fitting. Careful consideration of the shape of the visibility tunnel is essential. All factors like reflections, collimation, laser power density or small apertures (non-classic trajectories) should be taken into account.

The experiment should be repeated under modified conditions to eliminate any missed systematic errors and increase accuracy. Potential modifications include, among others:

• making both light paths equal in length,

• analyzing and/or avoiding interference caused by the single mode laser (fringes),

• a split of the beam in parallel towards more than 2 detectors,

• mounting apertures at each detector and testing the differences under varying configurations of reflection and beam split, e.g., 50/50 split,

• better ways to reduce diode noise, e.g., changing the wiring or moving the first stage amplifiers closer to photodiodes, even inside the vacuum chamber,

• more precise tuning of the wavelength to the absorption line using externally modulated laser driver, controlled with the standalone wavemeter or the lock-in feedback loop18,

• increasing the diameter of the vacuum chamber (extending the mean free path),

• more stable fastening of the multi-pass cell mirrors,

• removing the multi-pass cell rods shortening the mean free path near the laser beam,

• applying a different number of reflections in the multi-pass cell,

• assembling the measuring system outside the main section of the vacuum chamber so that the system could be reconfigured faster,

• testing other gases e.g., methane (also with a large cross section in NIR).

A slight modification of the proposed method will allow for conducting measurements of transmittance differences and mean free paths for larger working pressure ranges. In particular, the length of the light path should be adjusted by controlling the number of reflections in the multi-pass cell. The experiment can be carried out without the use of a multi-pass cell, using a sufficiently long vacuum chamber.

Another modification may involve enabling limiting and adjusting the mean free path by installing internal walls or pipes. This will allow for studying the relationship between the transmittance quotient and the mean free path.

The presented values of the transmittance quotient in a range of 1% may not seem respectable. However, higher values can be achieved e.g., by lengthen particles mean free time in a bigger chamber.

## Conclusions

For ultra thin gas we observed the effect of changing the optical transmittance measurement readings using detectors of different sizes. For detectors of a diameter much bigger than the wavelength transmittance increased as detector size decreased. Thus, the objective of the experiment, which is to observe in laboratory conditions the qualitative effect predicted by the non-local smeared gas transmittance model9, is achieved. The fact of observing a non-obvious phenomenon predicted by the theory is a strong premise for its validity. As far as we know there is no other gas transmittance model predicting such results.

The results are presented with a very high statistical significance level thanks to the long, repetitive measurement process. However, too few measurements were made to categorically determine the type of relationship between transmittance quotient and pressure. Moreover, there are many other improvements available regarding robustness, accuracy, flexibility etc. Some of them are outlined in the paper.

This experiment is a kind of a typical quantitative spectroscopy setup. It is relatively cheap and easy to repeat. However, it differs in two very important details from a well known TDLAS. Firstly, a sufficiently long gas particles mean free path is required. Free from any reactions that could be considered a quantum measurement or lead to decoherence. Therefore, the following should be provided: a large chamber, low pressure, electromagnetic shielding and weak measurement laser light. Secondly, the detector’s light sensitive diameter should be comparable (or lower) to the standard deviation of the smearing of the tested gas particles wave packets. It seems highly unlikely that both of these conditions could have been accidentally met during some earlier quantitative spectroscopy experiment. No reports of a similar experiment have been found in the literature.

In the experiment quite small detectors where used and a moderate pressure with a not so long light path was examined. However, the shown predictions of the smeared gas model apply for lower pressure levels, bigger detectors and longer distances. We believe the suitability of the smeared gas9 model should be verified in relation to the conditions of astronomical measurements.