Theoretical Research on Ellipsoidal Structure Methane Gas Detection Based on Near Infrared Light Sources of PbSe Quantum Dots

To improve the precision and sensitivity of the detection in near infrared gas detection system, the selection of light source and design of gas chamber structure are two key links. In this paper, the near infrared (NIR) light sources fabricated with PbSe quantum dots (QDs) and a new gas cell structure using an ellipsoid reflector were designed to test the concentration of methane (CH4). The double wavelengths differential detectionmethod was used in the paper.The signal wavelength is 1.665 μm from theNIRQD-based light source with 5.1 nm PbSe QDs. The reference wavelength is 1.943 μm from the NIR QD-based light source with 6.1 nm PbSe QDs. The experimental results show that the differential gain signal could be enhanced 80 times when the major axis, the focus, and the open length of the ellipsoid reflector are 4.18 cm, 3.98 cm, and 0.36 cm, respectively.The structure will be convenient for the signal amplifying, AD converting, and other process in the latter circuits, and therefore both the detection sensitivity and precision can be improved.


Introduction
Methane (CH 4 ) is a flammable and explosive gas.In recent years, great injuries and property losses were caused by frequent gas explosions, which has increasingly attracted the attention of the general public and the government.So, it has become an urgent problem to detect the concentration of methane immediately and accurately.In addition, methane is also a major greenhouse gas in the atmosphere, whose concentration needs to be detected exactly [1][2][3][4].
Existing methods for methane detection rely mostly on vector catalytic combustion, thermal conductivity, optical interferometry, and infrared absorption.Among various available methods, infrared spectroscopy absorption has its unique advantages over the other techniques [5,6].The sensing parts of infrared spectroscopy absorption gas detection system include an infrared launch system, a gas chamber, and a photoelectric conversion device.The selection of light source and design of gas chamber structure are two key links to improve the accuracy and sensitivity of detection.Recently, the emergence of the quantum cascade laser provides an alternate solution as infrared light source.However, quantum cascade laser as light source is expensive with complicated designs and structures, making it less desirable for the rising needs on portable, simple, and low-cost infrared gas sensors.
Semiconductor quantum dots (QDs) have been investigated and have shown very unique properties, such as the quantum confined optical property [7][8][9][10][11][12][13].They usually possess high photoluminescence (PL) quantum yield (QY) with size dependent tunable wavelength emissions, which makes them promising for light conversion [14][15][16].Among them, the PbSe bulk material has a small bandgap of 0.28 eV at room temperature and a very large exciton Bohr radius of 46 nm [17][18][19][20][21][22][23][24].As a result, PbSe QDs show very strong quantum confinement and high quantum yield in NIR region.Their band edge photoluminescence peaks span over a wide infrared wavelength region of 1−4 m [25][26][27][28][29].The wavelength can be adjusted merely by changing the particle size to cover the particular absorption frequencies of many kinds of gases in the NIR region.As a matter of fact, the NIR emitting QDs with narrow bandgaps have great application potential in gas detection.Compared to other infrared light sources for gas detection including infrared thermal emitter and semiconductor lasers, these NIR QDs have relative high modulate rate without the large thermal inertia, narrow emitting band without obvious interference, low cost, and small volume.
The normal gas cell in the infrared gas detection system can be divided into two categories: transmission type gas cell and reflecting type gas cell.When the narrowband light source is used to measure CH 4 concentration, the transmission type gas cell is more prone to produce coherent noise, which will affect the sensitivity of the system; the reflecting type gas chamber is more prone to produce interference noise, which can also bring a certain impact on the sensitivity of system [30][31][32].
In this paper, the NIR PbSe QDs and a new gas cell structure using an ellipsoid reflector were designed to analyze the concentration of CH 4 .Based on the double wavelengths differential detection method, the NIR PbSe QDs with the particle sizes of 5.1 and 6.1 nm are chosen as the NIR light sources.Due to the narrow PL spectra of NIR PbSe QDs, the filter needed in the conditional detection system can be omitted.Considering the low sensitivity of traditional gas cell structures, a new gas cell structure using an ellipsoid reflector is designed.The math model of the ellipsoid reflector is proposed.And the relationship between the parameters of the ellipsoid reflector and the light intensity of the detector is described.It can be proved that the structure can enhance the intensity of the output signal.

Establishment and Analysis of the Detection Model
In order to improve the sensitivity of infrared detection, a new gas cell structure using an ellipsoid reflector is designed, as shown in Figure 1.First of all, we establish the theoretical math model. and  are the semimajor axis and the semiminor axis of the ellipsoid reflector, respectively.As shown in Figure 1,  = √  2 −  2 and  = / < 1 and the focal point coordinates are  1 (+, 0) and  2 (−, 0).The signal light source  is placed at  1 ; the reference light source  is placed at a defocus position, and the detector is on  2 .
The NIR QDs light source can be treated as a cosine radiation.Radiation intensity in any direction  is where  0 (  ) is radiation intensity of wavelength   along axes.
According to Beer-Lambert law [33][34][35], the relationship between the reflected light radiation intensity and the incident light radiation intensity can be expressed as Ellipsoid reflector

Gas cell Out Input
Detector where (  ) is absorption cross section of CH 4 ;  is the concentration of CH 4 ; (  ) is ellipsoid reflectivity;  is the length of absorption path; (  ) is absorption coefficient of CH 4 .
The radiant flux received by detector can be written as where  0 is the distance between two focuses and  is the reception area of detector.The transmission distance  is a fixed value for the radiations of different angles.
The receiving model of detector can be written as where  0 is the response extent of detector.When   and   are adjacent,  0 =  0 .
The output electric signal converted by the detector is The output signal of detector without ellipsoid reflector is () can be seen as a constant for . Let In this design, the dual-wavelength differential absorption technology is employed.The signal wavelength   is 1.665 m from the NIR light source of 5.1 nm PbSe QDs.The reference wavelength   is 1.943 m from the NIR light source of 6.1 nm PbSe QDs.The NIR QDs lights are controlled in a time-sharing way.
Dividing ( 5) by ( 6), the differential signal can be obtained as However, the differential signal without ellipsoid is In ( 8) and ( 9),   is the output corresponding to   without ellipsoid reflector;    is the output corresponding to   with ellipsoid reflector;   is the output corresponding to   , which is almost constant no matter whether there is an ellipsoid reflector.
Therefore, after adding ellipsoidal structure, the gain of the differential signal is where Δ =  −  0 , representing the difference between reflected optical path through ellipsoid and direct incident optical path on the detector.

Experiment and Analysis
The absorption spectra and PL spectra of the NIR QDs light sources fabricated with 5.1 nm and 6.1 nm PbSe QDs are shown in Figures 2 and 3, respectively.The wavelength of 1.66 m is the central wavelength for CH 4 detection in the experiment.The near infrared absorption spectrum is shown  in Figure 4, which is from HITRAN atmospheric molecular spectroscopy database [36,37].
As seen in Figures 2 and 4, the PL peak of 5.1 nm PbSe QDs is 1665 nm with a full width at half-maximum of 143 nm, which covered the entire absorption spectrum of CH 4 gas.Carbon dioxide gas, carbon monoxide gas, and water steam are the interference gases in the experiment, whose absorption spectra are shown in Figures 5 and 6.As seen in Figure 5, the absorption coefficients of CO gas and CO 2 gas are small, so the interference influence can be ignored.It can be seen from Figure 6 that the overlap region between the absorption spectrum of H 2 O gas and the PL spectrum of PbSe QDs is very small, so the interference effect of water vapor can be considered very small.So, the NIR QDs light sources fabricated with 5.1 nm can be used to detect the concentration of CH 4 gas.
Figure 3 indicates the PL peak of 6.1 nm PbSe QDs is located on 1943 nm with a full width at half-maximum of 185 nm.As seen in Figures 2 and 3, the two wavelengths are adjacent.

Differential Signal Gains.
Refer to [30,32]; in the simulation computation, the related parameters are set as follows:  = 0.36 cm,  = 0.95,  = 2 mm × 2 mm, and  = 100 ppm, where  is the open length of ellipsoid reflector,  is the reflectivity of ellipsoid reflector,  is the effective reception area of detector, and  is the concentration of the CH 4 gas.In the ideal case, the different emulational curves of  and  with the increase of the semifocus  are shown in Figure 7.
As seen in Figure 7, the gain of signal relies on  and .When  is fixed, there is only one value of  corresponding to the maximum of gain.With the increase of ,  gradually increases up to maximum and then decreases.It is necessary to set a reasonable value for  and  of the ellipsoid reflector to get a suitable signal gain.In Figure 7, when we choose  = 1.99 cm,  = 2.092 cm, and  = 0.36 cm, the signal gain is approximately 593, which is enough to receive a strong signal.

Surface Curvature
Error.Surface curvature of the ellipsoid reflector has manufacture error.We choose  = 1.99 cm and  = 0.36 cm; the emulational curve of  and  with the fixed manufacture error  = 1 ∘ is shown in Figure 8.
As shown in Figure 8, when the manufacture error exists, the gain of the differential signal  will be greatly reduced.The maximum decreases from the original 593 to 80 times.This is mainly because that manufacture error will result in the reduction of  1 and  2 .Further, the reflected lights will not be received by the detector and the reflector surface of ellipsoid cannot be used entirely.The output power depends on the differential signal gain .The noisy power of the system is fixed; therefore, the improvement of the signal-to-noise ratio is and    ∝  2  ,   ∝  2  ; therefore, ΔSNR ∝  2 .In the system,  ∝ 10 1 , so the SNR was improved 10 2 .

Open Length.
As shown in (10), the open length of the ellipsoid reflector  will lead to some changes of  and , which will bring some impacts on the signal gain .After  and  are fixed, we simulate the relationship between gain  and  in Figure 9, where  = 1.99 cm and  = 2.092 cm.
As seen in Figure 9, when  and  are fixed, the gain of the differential signal  is monotonous index decreasing with the increasing of .

Conclusion
In this paper, the NIR light sources fabricated with PbSe QDs and a new gas cell structure using an ellipsoid reflector were designed to test the concentration of CH 4 .Based on the double wavelengths differential detection method, the math model of the ellipsoid reflector is established.And the relationship between the parameters of the reflector and the light intensity of the detector is described.It can be proved that the structure can enhance the intensity of the output signal.In simulation, based on the chosen parameters, the differential gain signal could be enhanced 80 times and the SNR could be improved 10 2 .In conclusion, the ellipsoid reflector structure will be convenient for the signal amplifying, AD converting, and other process in the latter circuits, which can be used to improve the detection sensitivity and precision in gas detection system.

Figure 6 :Figure 7 :
Figure 6: Normalized absorption spectra of PbSe QDs and absorption spectra of H 2 O near infrared.